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Fault Development Under Simple Shear: Experimental Studies

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Fault Development Under Simple Shear: Experimental Studies

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Content INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margin*;, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand corner and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back of the book. Photographs included in the original manuscript have been reproduced xerographically in this copy, Higher quality 6" x 9" black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. A Bell & Howell Information Company 300 North Z e e b Road. Ann Arbor. M l 48106-1346 USA 313/761-4700 800/521-0600 FAULT DEVELO PM ENT U N D ER SIMPLE SHEAR: EXPERIMENTAL STU D IES By Linji Y. An A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (Earth Sciences) May 1995 Copyright 1995 Linji Y. An UMI Number: 9614006 UMI Microform 9614006 Copyright 1996, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103 UNIVERSITY OF SOUTHERN CALIFORNIA THE GRADUATE SCHOOL UNIVERSITY PARK LOS ANGELES, CALIFORNIA 90007 This dissertation, w ritten by under the direction of h..i§....... D issertation Committee, and approved by all its members, has been presented to and accepted by The Graduate School, in partial fulfillm ent of re quirements for the degree of L inj i An D O CTO R OF PH ILOSO PH Y Dean o f Graduate Studies Date DISSERTATION COMMITTEE kIfLiu.- A H u Chairperson A ckn ow led gm en ts I have been worked closely with Professor Charles G. Sammis, my adviser, in the past five plus years. I always feel lucky, as all of his other students do, that I have gotten such a precious chance to work with a great man who has so many new ideas and so encouraging attitude toward his students. It was a great experience to be exposed to the new scientific frontiers such as fractals, chaos, self-organized criticality, and renormalization group theory. I appreciate all the encouragement and guidance from Charlie. I feel grateful to my committee members, Keiiti Aki, Gregory A. Davis, Iraj Ershaghi and Steve Lund for monitoring my progress and for their patience and time in reading my dissertation. The criticisms from these members helped in improving the overall quality of the dissertation. I especially benefited from discussions with Greg (Gregory A. Davis) on various topics in structural geology, and discussions with Professor Ershaghi on the application of my experimental results to fractured reservoirs. During the past six years at USC, many people come and go. Interaction with those people benefited my research and enriched my knowledge about the world. I got various helps from the faculty members in this department: Donn Gorsline and Robert Osborne (unfortunately he passed away in 1994) kindly gave me permission to get access to their lab facility, and Tom Henyey warmly lent me his instrument. The administration people, John McRaney, Rene Kirby, Kelley Virginia, Waite Cynthia, Denise Steiner and Desser Moton patiently helped me in handling various situations. Xiaofei Chen kindly answered my math questions and helped me locating the first apartment after I touched this land. David Bowman spent his precious time reading my manuscript and discussed fracture pattern development with me. He suggested term "protofracture" which I have formally adopted in my dissertation. Michelle Robertson and Jin Wang always lent me a ii hand when I was stocked in a computer problem, or when I forgot my office key (especially for Michelle). I had great time with my field trip fellows, Kim Bishop, Tom Brudos, Marry Park, Nick Taylor, David Mayo, Hacob Mkrtchian, Pawl Godin, John Bendixen, Kate Whidden, Hao Yu, Ken Fowler, Chris Carlson, Yang Zhang, Friedmann Julio, and Harold Ekstrom. Many discussions with Yong-Gang Li about fault structures were very inspiring. Chanting with Jinbo Chen, Lily Hsii, Periklis Beltas, Hongping Ouyang, Cong Wang, Xiaolin Liu, Inna Altchul, and Jiang Qu on various issues was existing. Playing tennis with David Adams was much fun. Carpool with Hongchun Li gave both of us convenience. I appreciate all these helps and good times. I also appreciate friendship developed during these years with Jin Wang, Zhen Lu, Chang-An Du, and Weishi Huang. It is definitely impossible to finish my degree without the consistent support from my family. My wife, Wenru You, always gave me understanding, encouragement and consistent support. It is hard to image to get any job down without her management of our family. I appreciate her patience, her concern, and her contribution to our family. I feel indebted to my daughter Rose An. I missed many times to attend her school meeting, and did not take enough care of her. I missed so much about my son Zhi An. He was a miracle bom in 1994 he is now in Beijing. I hope to get him back as soon as possible. I own my mother, Yulian Yang, a lot by leaving her so long without visiting her. She is always a spiritual support to me no matter where I am. I thank all of you for your patience, understanding, encouragement and concern. Table of Content Acknowledgment..................................................................................................... ii List of figures ....................................................................................................... viii Abstract................................................................................................................... xii Chapters 1 Introduction 1.1 Strike-slip faults in the crust................................................ 1 1.2 Previous studies of the structures of strike-slip faults 4 1.2.1 Field observations....................................................... 4 1.2.2 Theoretical studies..................................................... 6 1.2.3 Laboratory experiments............................................ 8 1.2.4 Numerical sim ulations........................................... 13 1.3 A new experimental technique for simple shear deform ation................................................................ 15 1.5 Organization of the dissertation......................................... 17 2 Experiments 2.1 Introduction........................................................................... 19 2.2 A pparatus.............................................................................. 19 2.3 Sample preparation and loading........................................ 21 2.4 Stress analysis of the experiment....................................... 22 2.4.1 Uniform simple shear............................................... 22 2.4.2 Torque.......................................................................... 26 2.5 Scaling of strain ra te ............................................................. 27 2.6 Monitoring of strain............................................................. 29 2.7 Monitoring of structure developm ent.............................. 30 3 Fault development in fine granular materials 3.1 Introduction 31 3.2 Sample preparation............................................................... 32 3.3 Experimental observations................................................... 33 3.3.1 Protofracture stage................................................... 33 3.3.2 Fracture stage............................................................ 44 3.3.3 Fault stage.................................................................. 47 3.3.4 Fault patterns............................................................ 50 3.3.5 Geometry of an individual shear structure 51 3.3.6 Evolution of fault steps and pull-apart basins 53 3.3.7 Fault rotation............................................................ 56 3.3.8 Scaling relationship between fracture propagation Rate and fracture length.......................................... 57 3.3.9 Scaling relationship between fault displacement And fault length........................................................ 59 3.3.10 Comparison of fault development in the three types of granular materials..................................... 60 3.4 Discussion................................................................................ 61 3.4.1 Implications of protofractures................................ 61 3.4.2 Why are major shear fractures and strike-slip faults not parallel to simple shear direction?................. 63 3.4.3 Comparison with Riedel shear studies................ 64 3.4.4 Comparison with natural fault patterns.............. 67 3.5 Summary................................................................................. 72 4 Faults development in a coarse granular material: Effect of material cohesion 4.1 Introduction............................................................................ 74 4.2 Experimental observations................................................... 75 4.2.1 Sample preparation.................................................... 75 4.2.2 Fault development in a coarse gouge layer 76 4.2.3 Comparison with fault development in fine granular materials.............................................. 79 4.3 Analysis................................................................................... 81 4.3.1 Relationship between particle size and cohesion .. 81 4.3.2 Displacement mode switch with particle size 85 v 4.3.3 Friction angle increase with particle siz e................ 86 4.3.4 Implications to fault development in the crust 92 4.4 Summary................................................................................. 93 5 Fracture displacement mode and internal fracture pressure 5.1 Introduction............................................................................ 95 5.2 Relationship between internal pressure and Stress intensity factors.......................................................... 98 5.2.1 Derivation of the expression for fcf......................... 98 5.2.2 Derivation of the expression for k u ..................... 101 5.3 Discussions............................................................................ 103 5.3.1 Variation of ki and ku during faulting................ 103 5.3.2 Implications............................................................. 109 5.4 A model for in-plane shear propagation of strike-slip faults in fine granular materials....................................... I ll 5.5 Summary............................................................................... 114 6 Origin of en echelon strike-slip faults 6.1 Introduction.......................................................................... 116 6.2 Experiments.......................................................................... 117 6.3 Experimental observations................................................. 119 6.4 Finite element analysis........................................................ 123 6.5 Models................................................................................... 129 6.6 Summary............................................................................ 137 7 A cellular automaton for the development of crustal Shear zones 7.1 Introduction.......................................................................... 138 7.2 A cellular automaton for the nucleation, growth Interaction of strike-slip faults......................................... 139 7.2.1 Nucleation of faults on the array......................... 140 7.2.2 Growth of faults...................................................... 141 7.2.3 Coalescence of faults............................................... 145 vi 7.2.4 Summary of automaton ru le s.............................. 149 7.3 Results and discussion........................................................ 149 7.3.1 Comparison with natural fault patterns.............. 156 7.3.2 Faulting param eters............................................... 166 7.4 Sum m ary.............................................................................. 180 8 Conclusions .................................................................................... 182 References.............................................................................................................. 186 vii L ist o f F ig u res Figure Title 1.1 Simple shear is pure shear plus rotation........................................ 8 1.2 The conventional representation of the assemblage of faults developed during a Riedel experiment......................................... 9 1.3 An outline of the coverage of the dissertation.............................. 17 2.1 The apparatus used in the simple shear experim ents................ 20 2.2 Forces acting on a sample layer subjected to gravity loading .... 22 2.3 Sketches showing the evolution of simple shear deformationfrom nonuniform to uniform ................................ 24 3.1 Fractures and faults developed in a less moist gouge layer during simple shear........................................................................ 34 3.2 Fractures and faults developed in a moister gouge layer during simple shear........................................................................ 37 3.3 Fractures and faults developed in a clay layer under simple shear...................................................................................... 40 3.4 Strike-slip faults terminate as protofractures, single shear fractures, en echelon shear arrays, or horse tail fractures 46 3.5 Sketches showing coalescence of fractures................................... 48 3.6 Primary shear fractures coalesce with conjugate primary shears and tensile fractures forming compound fractures.................... 49 3.7 Sketches showing two types of restraining bends observed in the experiments........................................................................... 53 3.8 Sketches showing the evolution of pull-apart basins observed in the experiments.......................................................... 54 3.9 Plot of initial fracture length Lj against the final length L2 of fractures developed in a clay layer................................................ 58 3.10 Fault displacement u is plotted as a function of length L .......... 59 3.11 Fracture assemblages developed under simple shear in this experimental stu d y ................................................................... 65 3.12 Fault pattern in southern California.............................................. 68 3.13 A Magellan radar image of the girded plain in the Guinnevere Planitia......................................................................... 71 4.1 Development of faults in a raw fault gouge layer under simple sh ear........................................................................... 79 4.2 Capillary pressure Pc is plotted as a function of particle size d...................................................................................... 84 4.3 A capillary in a granular material consists of pores and throats......................................................................................... 85 4.4 Simplified model of asperities with flat-top saw teeth geometry........................................................................................... 87 4.5 Variation of F /N with /J ............................................................... 88 4.6 Plot of bulk friction coefficient Pb versus asperity height h ...... 91 5.1 A simple shear experiment using fine fault gouge showing propagation of preexisting fractures............................. 96 5.2 A tensile fracture m odel.................................................................. 99 5.3 A shear fracture m odel.................................................................. 102 5.4 Plot of Ki* and Kn* as functions of normalized fracture length a /a o ...................................................................................... 104 ix 5.5 Effect of confining pressure on Kj* and Ku*.............................. 105 5.6 Effects of high fracture pressure and low confining pressure on Ki* and Ku*................................................................ 107 5.7 Effect of high confining pressure and low fracture pressure on Ki* and Ku*................................................................ 108 5.8 Effect of friction coefficient on Ki* and Kn*................................. 109 5.9 A suggested mechanism for in-plane propagation of shear fractures in granular m aterials.......................................... 112 6.1 Echelon shear arrays develop in a gouge layer............................ 120 6.2 An experiment using gouge layer with microslides inserted along the two optimum shear directions to represent preexisting shear fractures...................................... 121 6.3 Linear markers inscribed on a sample surface have been offset by primary shears and conjugate primary shears 124 6.4 Finite element analysis of shear fracture interaction................ 125 6.5 Mutual cutting of shear sets leads to the development of echelon shear arrays....................................................................... 130 6.6 A possible mechanism for the development of San Andreas -San Gabriel fault lo o p .................................................. 132 6.7 Echelon shear arrays developed mostly by fracture coalescence in a gouge layer.......................................................... 134 6.8 Sketch showing the development of echelon shears through fault coalescence............................................................. 135 6.9 Development of echelon shear arrays by mutual avoiding of collinear shears......................................................... 136 x 7.1 Sketches showing the orientations of faults developed under simple shear condition...................................................... 143 7.2 Stress state in front of a fault tip, and criteria for fault interaction....................................................................................... 147 7.3 Concept of two strike-slip fault interaction over their process zones................................................................................................ 148 7.4 Evolution of a simulated fault pattern........................................ 150 7.5 Fault pattern in the Mojave Desert in southern California .... 157 7.6 The fractal dimension of a through-going shear z o n e 159 7.7 Length distribution of faults in an automaton-generated fault pattern.................................................................................... 161 7.8 Length distribution of shear segm ents......................................... 163 7.9 Plot of normalized segment length vs. normalized displacement for natural strike-slip faults................................. 164 7.10 Arrangements of fault segments in simulated and natural strike-slip faults.............................................................................. 165 7.11 Fault patterns generated with different R 's ................................ 167 7.12 Effects of R on Df, dc and m ............................................................ 169 7.13 Effect of velocity exponent n on fault patterns........................... 171 7.14 Fault patterns generated with different B 's................................. 174 7.15 Effect of B on Df, dc and m .............................................................. 178 7.16 Effect of array size on on Df, dc and m ......................................... 179 ABSTRACT Simple shear experiments using moist clay and fault gouge layers revealed that a strike-slip fault nucleates as protofractures and develops as the protofractures grow into shear fractures and then the shear fractures coalesce. Most coalescence occurs by taking advantage of preexisting structures. Fault coalescence and interaction create three types of releasing bends and two types of restraining bends. A fault pattern in a broad shear zone consists of several conjugate fault sets. The most developed set contains nearly equal-spaced, parallel strike-slip faults. This set establishes a structural framework within which all the other sets are developed. None of the major faults is parallel to the applied simple shear direction due to internal friction. All the shear structures propagate in-plane. Conditions for shear propagation of fractures are explored through experiments and theoretical analysis. A critical condition for shear propagation is a lower ratio of fracture pressure versus confining pressure. If the ratio is much lower than one, shear fractures propagate in-plane. If the ratio is close to one or higher, shear fractures nucleate out-of-plane tensile fractures. A small ratio can occur if a fracture is sealed. Small particle sizes in a granular material, appropriate water content, higher confining pressure and material cohesion all contribute to fracture sealing, and promote shear extension. It is thus suggested that shear fracturing should be more common at depth than in the shallow part of the crust. Three mechanisms are proposed for the origin of echelon shear structures. These are mutual shearing of conjugate shear structures, coalescence of discrete shear structures, and mutual avoiding of collinear shears due to conjugate shearing, ductile deformation, or transient tensile fracturing. The experiments and computer simulations allow the constraint of several faulting parameters. Linear relationships are found between average fracture propagation rate and fracture length, between the maximum jump distance of a shear structure and the combined length of the two interacting structures, and between the displacement of a strike-slip fault and its length. Chapter 1 INTRODUCTION 1.1 Strike-slip faults in the crust Major strike-slip faults of the world develop along transform plate boundaries under simple shear condition (Luyendyk et al. 1985, Woodcock 1986, Sylvester 1988). A large strike-slip fault usually does not develop alone, but appears as one element of a complicated fault system in a shear belt. A shear belt contains several fault sets of different orientations and mechanical properties. Each fault set includes many parallel or subparallel faults of the same mechanical properties. In a shear belt, the largest strike-slip fault spans the whole region, and accommodates most shear strain in the region. Such a fault is called a through-going shear zone. All the other smaller faults belonging to different fault sets cross over and intersect each other, forming an integrated fault network. In such a fault network, any event (e.g., earthquake) on one fault can cause stress readjustment on all the other fault, leading to a chain reaction (Bak and Tang 1989, Huang and Turcotte 1992). Aftershocks following a major event are typical examples. Intensive studies of the detailed structures of strike-slip faults have been motivated by observations that strike-slip faults are the most important earthquakes-generating structures (Allen 1981). The most recent devastating earthquake in Japan (M=7.2) on January 17, 1995, occurred right on a blind strike-slip fault. Paleoseismic investigations imply that earthquakes occur more frequently on strike-slip faults than on intraplate normal and reverse 1 faults (Sylvester 1988). Beside their importance in earthquake study, strike- slip faults are also one of the three fundamental elements of plate kinematics (Woodcock 1986). They adjust stress between different plates and help continent accretion. Large displacements along major strike-slip faults truncated continents and caused thousands of kilometers of terrain traveling in geological time (Davis et al. 1978, Jarrard 1986). It has been found that at least 50 terrains in western North America and Alaska are allochthonous and most of which are separated by large strike-slip faults (Jarrard 1986). Strike- slip faults and shear fractures are also associated with mineral deposits. It is well known that displacement on a large strike-slip fault creates pull-apart basins and en echelon folds which can become petroleum traps (Harding 1974, Dibblee 1974). The fractured formation can become important reservoir rock. Monterey formation which is an important reservoir rock in California is a typical example. Fractures in a formation create both conduits for fluid flow and space for fluid storage. The production of a fractured reservoir depends on the density of fractures, fracture apertures, connectivity, and attitude. Steam entries at The Geysers in northern California are produced exclusively from a relatively small number of subhorizontal major fractures (Thompson and Gunderson 1989). A complete understanding of a strike-slip fault system must include its geometry (fault length, width, depth, strike, dip, steps, connection, overlap, branching, curvature and fault surface roughness, e.g., Naylor et al. 1986, Aydin and Schultz 1990, Scholz et al. 1993), kinematics (displacement, shear sense, strain, and ductile flow, e.g., Crowell 1962, Davis and Burchfiel 1973, Ramsay 1980) and dynamics (nucleation, propagation path, mode and rate, interaction, friction within a fault and slip stability, stress, and stress drop, e.g., 2 Rice 1968, Segall and Pollard 1980, Du and Aydin 1991, Segall and Pollard 1983, Reches and Lockner 1994). However most previous studies focus on an individual fault or a pair of individual faults, and the fault pattern as a whole has rarely been studied. Little is known about multi-fault interactions, coalescence, branching, the origin of fault discontinuities, the effects of confine pressure and internal fault pressure, and the brittle-ductile crustal coupling. Many controversies about strike-slip faults, e.g., if a strike-slip fault can propagate in-plane (Melin 1986, Petit and Barquins 1988, Cox and Scholz 1988), and what kind of fault steps, releasing or restraining, is easier for rupture propagation (Segall and Pollard 1980, Sibson 1986), remain unsolved. The reason so many questions remain unanswered is that there is no simple shear technique to study the evolution of a strike-slip fault system in laboratory. As elaborated in the next section, field observations are limited in time, theoretical studies are too difficult, and previous laboratory experiments have had restrictive boundary conditions. The principal objective of this dissertation is to develop a new laboratory technique to study simple shear deformation in laboratory. This new technique is significantly different from the traditional Riedel shear technique which is believed inappropriate for regional shear belt studies, as will be discussed below. The technique is then used to study the nucleation, propagation, and interaction of strike-slip faults in a broad region leading to the development of a through-going shear zone. Specifically the following problems will be addressed: 1) fault nucleation: where and how a fault nucleates; 2) propagation: does a strike-slip fault propagate in plane or out of plane? How is a fault displacement mode affected by confining pressure, fault internal pressure, and material cohesion? 3) interaction of faults: when and how strike-slip faults interact and coalesce; 4) 3 the origin of fault discontinuities: why does a strike-slip fault consist of en echelon segments? and how do fault segments step over? 5) the orientation of strike-slip fault: should a major strike-slip fault be parallel to the regional simple shear as is usually assumed? 6) earthquake barriers and pull-apart basins: where do they occur and how do they evolve? 7) fault pattern: what kind of fault pattern develops under simple shear in general? 1.2 Previous studies of the structures of strike-slip faults In this section, I review previous field observations, theoretical and laboratory studies of strike-slip faults and shear fractures. 1.2.1 Field observations In field, several sets of strike-slip faults can be identified within a shear belt. The most developed set is the one that is about parallel to the direction of the shear belt. A conjugate set to this fault set can also be found which is about perpendicular to the shear belt extension and is generally much weaker. Strike-slip faults within each fault set are parallel or subparallel (Sylvester 1988). Each individual strike-slip fault is discontinuous, consisting of many segments (Segall and Pollard 1980, Segall and Pollard 1983). Different segments are either linked up or totally separated. The linkage of two non- coplanar segments forms a step. Such a step is referred to as a releasing bend if the stress concentration on the step is extentional, and restraining bend if the stress concentration is compressional (Crowell 1974). Further displacement on a strike-slip fault turns a releasing bend into a pull-apart 4 basin while a restraining bend, an up-warp (Segall and Pollard 1980, Deng et al. 1986). A basin developed in such a way has been observed to have a width/length ratio of -1 /3 (Aydin and Nur 1982). Strike-slip faults have been observed nucleating from preexisting tensile fractures and joints (Pollard and Segall 1983), and terminated as splays (Simpson 1983), brush structures (Segall and Pollard 1983), and tensile fracture arrays (Liu 1983, Granier 1985). The vertical structures of strike-slip faults have been studied from deeply eroded crustal shear zones (Sibson 1977, Anderson 1983, An 1991). A general picture is that, the upper-most part of a strike-slip fault (above 4 km) may be flower-structured in cross-section, the middle part (4 km to the brittle/ductile transition boundary at 10-15 km depth) becomes a single discontinuity, and the lower part (lower than the transition boundary) becomes wide again and deformation becomes totally ductile. Discontinuous fault surfaces disappear at this depth. Sibson proposes (1977) the brittle/ductile transition boundary is a seismogenic zone because of high stress concentration. The stress state of a strike-slip fault is more complicated than simply shear. Some strike-slip faults have more compressional component (e.g., San Andreas) while others have more dilational component (e.g., Red Sea rift zone, see Mount and Suppe 1992). In a single fault, stress condition varies in different segments. In situ stress measurement on San Andreas, Sumatra, Philippine, and Alpine faults indicate that current maximum principal stresses are almost perpendicular to the fault traces while on the Kane and Dead Sea transforms, the minimum principal stresses are oriented at a high angle to the fault traces (Zoback et al. 1987, Mount and Suppe 1992). 5 Earthquakes occur not only at the propagating tips but also in between. Earthquakes occur within a strike-slip fault are related to asperities (Aki 1979, Rudnicki and Kanamori 1981) or barriers (Das and Aki 1977, Aki 1979). The asperities are closely related to fault geometry in that a strike-slip fault is not smooth linear structure but contains jogs or bends (Crowell 1974, Segall and Pollard 1980). Observations indicate that an earthquake does not generally rupture the whole length of a strike-slip fault but only a fraction of it (Wesnousky 1989). The endpoints of strike-slip earthquake ruptures are commonly associated with steps along a fault trace (Segall and Pollard 1980, Sibson 1986). The stress concentration at a step is predicted to be a function of the width of the step as measured perpendicular to fault strike (Segall and Pollard 1980). Earthquake rupture propagation indicates that the stress concentrations at fault steps may be sufficient to impede rupture propagation. Therefore there may be a coincidence between the endpoints of earthquake ruptures and the location of steps along a fault trace (Sibson 1986, Barka and Kadinsky-Code 1988, Wesnousky 1989). If it is so, fault geometry, especially the origin and evolution of fault steps, becomes a key in understanding earthquake process. 1.2.2 Theoretical studies Linear elastic analysis treats a strike-slip fault as a mode II (in-plane shear) or mode HI (anti-plane shear) crack (Rice 1968, Segall and Pollard 1980, Pollard and Segall 1987, Li 1987). Analytic expressions for stress field and displacement field of a single crack in homogeneous and isotropic media have been derived for pure shear stress configuration (Rice 1968, Pollard and 6 Segall 1987). Stress concentration at a crack tip has been studied by ways of stress intensity factor, energy release rate, and J-integral (Ewalds and Wanhill 1984, Li 1987). A crack propagates when a critical value of one of the parameters is reached. In such studies, friction on fault surfaces is not generally considered and a shear crack is assumed to develop exactly along the maximum shear plane. Interaction of two shear cracks has become a topic of many recent studies (Segall and Pollard 1980, Chen 1984, Kachanov 1987, Du and Aydin 1991). Du and Aydin (1993) concluded that no matter what is the shear sense and fault step, the propagation paths of two en echelon mode II cracks always converge toward each other. Theoretical stress fields for two cracks interacting in an elastic homogeneous media have also been given by many studies using the principle of superposition (Chen 1984, Kachanov 1987), method of successive approximation (also called Schwarz alternating technique, see e.g., Segall and Pollard 1980, Aydin and Schultz 1990) and method of asymptotic approximation (Chang 1982, Du and Aydin 1991). Study on the interaction of more than two cracks has not been found in literature. A debate in fault step studies is which step, releasing or restraining step, constitutes a barrier for rupture propagation. Segall and Pollard (1980) proposed, based on their 2-D quasi-static study of shear cracks, that since compressional steps cause an increase in the mean stress and an increase in the normal stress, compressional steps should act as barriers to rupture propagation. However, Sibson (1986) argued that dilational steps should act to stop earthquakes because the tensile regime created by a dilational step should act to momentarily decrease pore-fluid pressure. The decrease of fluid 7 pressure would cause an increase in fault strength and thereby stop earthquake rupture. A more challenging problem in fault geometry study is the dynamic aspects of asperities: the number, location, and geometry of asperities always change with fault displacement. Some asperities are destroyed while others are created. If this dynamic process can be simulated by an experiment, then the chaotic earthquake nucleation will become more transparent, and we will have a more robust physical base for earthquake prediction. 1.2.3 Laboratory experiments Shear deformation is usually divided into simple shear and pure shear. Simple shear and pure shear differ in their stress configurations. In pure shear, compression stress is applied to a sample, and shear structures develop along planes on which the derived shear stress is maximum (theoretically). Pure shear is coaxial (the principal stress axes do not rotate during progressive deformation). A majority of shear experiments with rock specimens have been performed in this manner. Simple shear is a stress configuration in which shear stress is directly applied on a sample. Simple shear corresponds to pure shear plus rotation (Means 1976, see Fig. 1.1). Since principal stress axes keep rotating with progressive strain, the deformation is non-coaxial. The non-coaxial deformation is observed in nature, especially between plate boundaries, but it is not easy to simulate in laboratory. The primary problem is that the not- coaxial rotation of a specimen during a simple shear experiment is 8 simple shear (b) pure shear (a) Fig. 1.1 Simple shear is pure shear plus rotation, (a) shows pure shear and (b) shows simple shear. From (a) to (b) the object rotated an angle a . accompanied by the variation of sample geometry which makes it difficult to maintain a stable simple shear boundary condition. The experiments that are analogous to simple shear can be divided into three types. The first one is called Riedel shear (Cloos 1928, Riedel 1929). In this experiment, a slab of clay is placed horizontally over two parallel adjacent boards. One board is then slid horizontally past the other. Faults and sometimes folds develop in the overlying clay layer (Fig. 1.2). The first structures to form are a conjugate pair of synthetic and antithetical shear faults known as Riedel shear and conjugate Riedel shear (R and R' in Fig. 1.2). These are followed by secondary synthetic shear faults denoted by P. Another secondary antithetical fault, X fault, which is conjugate to P fault was added by Bartlett et al. (1981). However, Naylor et al. (1986) argued that the X structures may have been rotated primary antithetic faults R'. The real conjugate fault to P should be P' which is between primary conjugates R and R'. Neither P nor X has been conclusively documented. 9 t PDZ Fig. 1.2 The conventional representation of the assemblage of faults developed during a Riedel experiment. PDZ: principal deformation The Riedel experiment was later extended to study more complex structural features. Some experiments incorporated components of divergence and convergence during shear deformation (Wilcox et al. 1973, Naylor et al. 1986, Withjack and Jamison 1986, Smith and Durney 1992). They showed variations in the orientation of structures from the Riedel model. Tchalenko (1970) performed experiments using both engineering shear box and the Riedel configuration to demonstrate the influence of shear strength on the development of structure and the similarity between the experimental results and natural deformation. These experimental results have been shown to be applicable directly to rock by Bartlett et al. (1981) by deforming veneers of limestone. zone. 10 Both the Riedel and modified Riedel experiments assume a preexisting fault underneath a cover layer. The reactivation of the old preexisting fault in the basement causes the deformation in the cover layer. The structures developed in this manner are therefore secondary which are not indicative of the origin and development history of the underlying master fault. The experiment considers only one master fault, which is not suitable for studying multiple-fault interaction. The second type of experiments is the ring-shear experiment performed by Mandl et al. (1977). They used an apparatus consisting of a transparent annulus, and the lower part of which can be slowly rotated but the upper part is fixed. Granular materials were filled into the annulus and sheared horizontally. They observed development of shear bands and dilation during shearing. Although the samples were sheared from their initial "fault-free" condition, and shear stresses were applied by taking advantage of sample weight which advanced the technique in simple shear experiment, it is not proved, either by theoretical analysis or by the experimental results, that stress condition in this experiment is indeed uniform simple shear. One obvious problem of the technique is found in the boundary condition. A sample is bounded all around by the annulus. The friction or cohesion force between the sample and the boundaries can not thus be ignored. If so, the stress state of the sample may be complicated and cannot be uniformly simple shear. The third type is friction experiment in which a layer of fault gouge, usually artificial one, is sandwiched between two saw-cut surfaces and the sample is deformed under pure shear condition (Dieterich 1979, Marone et al. 1989, Biegel et al. 1989, Morrow and Byerlee 1989, Marone and Kilgore 1993). 1 1 The sliding of the rock surface with each other applies a shear stress on the gouge layer. Beside the friction property, textures developed within the gouge layer are sometimes examined for structure and strain evolution. As in Riedel shear, the 'fault' is preexisting, and thus it can not be used to study fault nucleation. The shear is restrained within a narrow band of fault gouge, and the sample within the narrow band undergoes processes more like 'crushing' rather than deformation in common sense. The experiment is generally performed under high strain rate which may greatly affect sample's mechanical behavior, and the deformation process cannot be seen until the experiment is over when unloading recovery has taken place. Most of all, the experiment is pure shear and shear strain is limited. These indicate that the experiment is not suitable for the study of regional strike-slip fault pattern. An important debate about strike-slip fault development is whether it can extend in its own plane. Since strike-slip faults, some of them can be thousand of kilometers long, are linear structures which cause significant offset along the fault traces, it seems to be a straight forward conclusion that a strike-slip fault develops by in-plane propagation. This conclusion is also backed by Lawn and Wilshaw (1975) and Melin (1986) by their theoretical analysis which concluded that high confining pressure promotes shear displacement. However, laboratory experiments with rock specimens all failed to produce in-plane shear propagation (Brace and Bombolakis 1963, Lajtai 1971, Ingraffea 1981, Shamina et al. 1975, Horii & Nasser 1985, Sammis and Ashby 1986, Ashby and Hallam 1986, Cox and Scholz 1988, Petit and Barquins 1988). What they observed is that a crack loaded in shear, or with combined normal and shear loads, propagates out of its plane, curving into 1 2 principal plane parallel to the maximum compressive stress. In order to use this experiment to explain the development of strike-slip faults, they developed an idea which considers that before any macroscopic shear surface appears, tensile microcrack arrays must develop first in front of a crack tip. The further damage of the cracked region leads to a macroscopic shear extension (Cox and Scholz 1988, Petit and Barquins 1988, Reches and Lockner 1994). Petit and Barquins (1988) further ruled out mode II (shear) as a preliminary crack mechanism based on such an experiment observation. 1.2.4. Numerical simulation In recent years, numerical simulation has become an important trend in studying fault process. A number of techniques have been developed for this purpose. In studying earthquake nucleation and the relationship between earthquake population and magnitude (so-called Gutenberg-Richter law), linear elastic (spring) model is used (Beale and Srolovitz 1988, Lockner and Madden 1991a, b, Godano 1991). In deriving the stress state along a fault and dealing with fault interaction, approximation methods such as finite element (Ingraffea 1987), finite difference (Harris and Day 1993) and boundary element methods (Crouch and Starfield 1983) are used. In studying fault pattern evolution, probability models such as branching model (Vere-Jones 1977, Watanabe 1986), percolation model (Watanabe 1986, Sahimi 1993, Su and Yan 1993), Marcov chain model (Gansted et al. 1991) and cellular automaton (Steacy and Sammis 1992) are adopted. These methods focus on different aspects of fault problem, and the results are quite assumption- dependent. At this time, satisfactory results can only be derived for the stress 1 3 state of a single or a couple of cracks. When more cracks are involved, the stress distortion caused by those discontinuities soon becomes too complex to solve, and computation also becomes extremely intensive. Because of this, heterogeneity problem cannot generally be included in the simulation, and coupling force between different tectonic layers of the crust has also been ignored. However, more completed and sophisticated software are coming out, and it is expected that it may become a powerful tool in the future in solving faulting problem. A quite different numerical simulation technique should be mentioned which is a 3-D fracture model used in engineering for reservoir simulation. It is a simple dual porosity model proposed by Warren and Root (1963). Large fractures in the model are simplified into three sets of perpendicular and regularly spaced arrays which partition the reservoir into cubic "matrix" blocks. The net storage and transport contribution from all the smaller fractures in the reservoir are lumped into average values for the porosity and permeability of these matrix blocks which then feed the large fractures. The model was proposed when insufficient data about fracture density, orientations, dip, aperture, connectivity, and conductivity are available. Production practice at The Geysers indicates that steam entries often occur in clusters in certain orientations, and these most productive "backbone" fractures seem to have more or less tensile component (Thompson and Gunderson 1989). Field observations indicate that a fracture system may be fractal rather than three perpendicular, regularly spaced discontinuities (Sammis et al. 1992). The Warren and Root’ s model (1962) therefore need to be modified to incorporate these observations for more accurate simulation. 1.3 A new experimental technique for simple shear deformation Above review indicate that true simple shear experimental technique is not mature yet in laboratory. The Riedel shear uses a preexisting "fault" as a boundary condition to generate new faults. It is only suitable for studying secondary structures related to a preexisting fault underneath. Other rock experiments with sawcuts have been performed under pure shear condition instead of simple shear. In the experimental studies mentioned above, the coupling force between the upper brittle and lower ductile crustal layers is ignored. However this boundary condition is critical for stabilizing fault propagation and generates earthquakes in the crust. In a typical laboratory experiment or theoretical analysis, only a few (usually a single or a couple) faults are considered. This is the simplest case if the result is to be used to explain shear zone evolution which involves growth and interaction of a multitude of faults of different orientations. It is made even ideal by using intact materials without considering preexisting structures. A natural strike- slip fault, as observed, always develops within the context of a fault pattern in which several fault sets exist, and fault interaction is multiple. The last, may be the most difficult, problem is about fault network. Faults within a shear belt coalesce and crossover forming a self-organized, may be critical, fault network (Burridge and Knopoff 1967, Bak and Tang 1989). On such a network any event, e.g., an earthquake, causes stress readjustment in all the other locations and chain reaction. Earthquake prediction requires knowledge about where new stress concentration will occur, how fast it can accumulate, and where new faults will grow. All of these require understanding the evolution of an entire fault pattern, and this understanding can be better approached by experiments for reasons that geological time scale is too long for a complete observation of stress accumulation and theoretical analysis is not practical at this time. In this dissertation, a new experimental technique has been developed to meet these needs. True simple shear has been achieved in a broad region by taking advantage of gravity body force. Large sample layers made of clay and fault gouge has been used to give enough space for fault propagation and interaction which lead to the development of a complicated through-going shear zone. The granular materials, clay and fault gouge, have been used because it is believed that the brittle crust is more like granular materials than coherent elastic continuum (Gallagher 1981, Scott et al. 1993), and because the granular materials are more ductile which permit a slow fault propagation that is otherwise too fast to observe in elastic media. Using this technique the coupling force between the brittle and ductile layer of the crust can be simulated by the frictional force between the sample layer and the loading base. With this technique the whole process of fault development from nucleation, propagation to interaction can be observed from the very beginning to the end. The experiments demonstrated convincingly that a strike-slip fault does not develop parallel to the simple shear as it is usually assumed, a shear fracture can propagate in plane under certain conditions, both releasing and restraining bends can lead to the development of pull- apart basins, fault discontinuities are related to conjugate shearing, and fault coalescence takes advantage of preexisting faults. 1.4 Organization of the dissertation An outline of the dissertation is shown in Fig. 1.3. Following this introduction, Chapter 2 gives a description of a new technique for simple shear experiment. In Chapter 3, experiments on fault development in granular materials (clay and fault gouge layers) are presented. From Chapter 4 to Chapter 6 are detailed studies of some aspects of strike-slip faults. Chapter 4 develops the idea that material cohesion has a similar effect on fault development as does confining pressure. Chapter 5 studies the dependence of fault displacement mode on both confining pressure and fault internal pressure. Chapter 6 explores the origin of en echelon shear arrays commonly observed in both laboratory experiments and in field. Chapter 7 puts all the observed fault parameters to work in a computer automaton which is used to explore the effect of fault parameters on the emergent fault pattern. Chapter 8 summarizes the results of these studies. ♦simple shear experiments nucleation propagation orientation displacement mode { r mteraction [♦effect of confining pressure versus fault pressure ♦effect of material cohesion propagation rate termination interaction geometry bridging structures interaction styles fault sets & orientations ♦ origin of fault discontinuities patterns fault spacing fault trace geometry asperities and pull - apart basins ♦ indicates independent chapters ♦automaton simulation Fig. 1.3 An outline of the dissertation structures Chapter 2 EXPERIMENT 2.1 Introduction The experiment is designed to study fault pattern development in 2-D. The crust is therefore treated as one layer with limited thickness but very large lateral extension. The deformation in the crust is assumed to be driven by mantle convection. The coupling between the brittle upper crust and the ductile lower crust has been simulated by the frictional force between the layer and its base. Strain rate (loading rate) has been chosen to be appropriate to model the development of natural strike-slip faults in a broad zone subjecting to simple shear, such as along transform plate boundaries. The experiment can demonstrate, step by step, the nucleation, growth, interaction and development of through-going shear zones. This experiment is different from traditional Riedel shear (Cloos 1928, Riedel 1929) in that it does not have a preexisting master fault (therefore all structures are primary), the simple shear is uniform throughout the cake, and the coupling between the brittle and ductile crustal layers can also be simulated. 2.2 Apparatus The apparatus is shown in Fig. 2.1a. It is composed of an aluminum board (60 cm xl20 cm) with two raised edges. The board is used to support clay and gouge layers. The raised edges are used to fix one of the sample 19 sample sam ple plastic wrap i ^ — i \ i s ' \ aluminum board w ater film (b) Fig. 2.1 a). The apparatus used to perform simple shear experiment consists of an aluminum board (1), a holder (2), a pulley (3), a lock pulley (4), and a piece of rope (5). b). A cross section showing sample loading procedure. 20 boundaries. This board is fixed to an axis about which it could be tilted up to 90°. A rope and pulley system are used to maintain and adjust the tilt angle of the board during an experiment. 2.3 Samples preparation and loading Clay and fault gouge from the San Andreas and the San Gabriel faults were used to make the experimental layers. Fault gouge was used because it is less cohesive and more brittle than clay, and the cohesion can be controlled by adjusting its grain size. The materials were mixed with ~39-41wt.% of water depending on different purposes: the materials with higher water content were used to achieve large shear strain (can be as large as 1.5) while the materials with lower water content were used for well developed shear structure (less than 1). Fault structures also differ with water content in the same material. Test showed that this water content gives the materials ability to transmit stress during deformation, and yet soft enough to deform under gravity. A sample layer was loaded following the technique illustrated in Fig. 2.1b. First, the board was set in a horizontal position. Then it was moistened with water and a layer of plastic wrap was laid on the surface. The purpose of the plastic wrap was to reduce friction between the board and the clay or gouge layer. Only the surface of the board, not the raised edge, was covered with plastic, so that the layer can slide down on the plastic covered surface upon tilting while one of its boundary remained was attached to the uncovered edge, thereby generating a simple shear. A thin film of water was spread on the plastic wrap to reduce the friction further. In some experiments 21 we tried corn oil lubricate. It turned out that the oil reduced interface friction so much that samples did not deform in simple shear but rotated on the board and then tore off the fixed edge. Therefore an appropriate interface friction is necessary which can be used to simulate the coupling force between the upper brittle crust and the lower ductile crust. Finally, the clay (or gouge) layer was placed on top of this water film. The dimension of the sample layer was configured to be 80x40x2.3 cm3 . 2.4 Stress analysis of the experiment 2.4.1 Uniform simple shear A s an experiment starts by tilting the board, two forces are applied on the sample simultaneously (Fig. 2.2). One is gravity body force and the other is basal friction. The gravity body force can be resolved into slope-parallel and normal components. The magnitude of the parallel component Fp is Fp -m g sin a , (2.1) Fig. 2.2 Forces acting on a sample layer subjected to gravity loading 22 and the magnitude of normal component is Fn = mg cos a (2.2) where p is the density of the materials, m is the mass, g is gravity, and a is the tilt angle of the board. The Fp makes the sample slide down the slope while Fn contributes to the basal friction Ff : where p is the coefficient of friction, AP is the difference between confining pressure P and pore pressure Pp (AP=P-Pp), A is the area of the sample, and So is the inherent shear strength of the surface. In order to understand the mechanism of uniform simple shear, let us assume that a sample can be divided into numerous rectangular mini-zones as shown in Fig. 2.3a. Because sample thickness is the same everywhere, the slope-parallel component of the gravity force within each minizone is the same, i.e., Fl=F2=F3=...=Fn. If no boundary of the sample is fixed, all the minizones will move at the same speed and no strain can be induced. However if one boundary is fixed as in Fig. 2.3, then the minizone closest to the fixed edge (minizone 1) develops shear strain first. The shear strain causes strain hardening within the minizone. The strain hardening can be measured by shear force induced in the minizone. The magnitude of the shear force in an elastic material is where A* is the area of the lateral cross section of the sample, G is shear modules, and y is shear strain. In a moist granular material, the relationship between Fs and y may be more complicated but Fs should still increase with y . Ff =-Ui(FK+AAP) + AS0] (2.3) Ff =A*Gy (2.4) 23 f=S fixed edge I — fixed edge F1=F2=F3=F4 7l>72>y3>y4 (a) F1=F2=F3=F4 7l=72^y3=74 (b) Fig. 2.3 Sketch showing deformation evolves from initial nonuniform in (a) to finally uniform simple shear in (b). Deformation in each small box in (a) and (b) is enlarged in the lower parts. Fn and yn denote gravity force and shear stress in n'th zone. Shadows show the extent of strain. 24 Strain hardening along can not explain the development of uniform simple shear in the granular samples because when corn oil is used as lubricate, a sample rotates and fails along a narrow localized shear zone near the fixed edge. Therefore the second important condition to achieve uniform simple shear is the increase of basal friction with shear strain. As showing in Fig. 2.3a, when one minizone is more strained (e.g., minizone 1) than another, the dilation of the minizone causes pore pressure to drop within the minizone (Ap to increase in Eqn. 2.3). Pore pressure drop causes interfacial water to be sucked up by the pores and thus the friction between the sample and the base is increased. At the same time, pore pressure drop is equivalent to an increase in normal stress. According to Eqn. (2.3), increase in AP also leads to an increase in basal friction. Basal friction plays at least three important roles: resisting rotation accompanying simple shear, preventing sample acceleration, and inducing shear strain by resisting further deformation in the more strained minizones. When a more strained minizone (e.g., minizone 1) slows down due to increase in basal friction, it works as another fixed edge to induce shear strain in the adjacent minizone (e.g., minizone 2), and then the minizone 2 again slows down and works as a fixed edge to induce shear strain in the minizone 3. This process will continue until the last minizone is deformed. A uniform shear deformation is thus developed (Fig. 2.3b). The subsequent deformation is maintained uniform simple shear by similar mechanisms. Because of strain hardening and the increase in basal friction, the sample has to be tilted more (to increase gravity loading) to maintain a constant deformation rate. For each small increment of gravity loading, the forces acting on each minizone will adjust to establish a new 25 equilibrium between driving force and resisting forces. Enough time is need for the minizones to adjust from one state to another. Therefore deformation rate should be slow. 2.4.2 Torque Friction not only helps in realizing uniform simple shear, but also balancing torque which causes rotation of a sample. The torque for a rectangular sample layer changes with progressive deformation. It is the maximum at the beginning of a deformation. Simplify the layer as a beam of the same weight with mass distributed uniformly along the beam, the torque where p is the density of the layer, g is gravity, /, zv, and h are the length, width and thickness of the layer, respectively, a is tilt angle, and < p is the angle between rotation and gravity sliding directions. If this torque is not balanced by a force, the sample layer will rotate on the board. This was the case when oil was used as lubricate. To prevent this from happening, the torque must be balanced by a force, say, friction: The variables on the right-hand are p., Fn and AP. Because increasing Fn increases M and increasing AP decreases F„, the only efficient way to make M<-Ff is to choose a lubricate with large p . Tests shows that water can work well as lubricate for this purpose. M is; M = pgwh sin a cos <p\rdr = — pgwhl2 sin a cos (p (2.5) M < -F f =p{Fn + A&P) + AS0 (2.6) 26 2.5 Scaling of Strain Rate Strain rate greatly affects the property of a material. A brittle material (e.g., rock) becomes ductile under very slow strain rate, and a ductile material (e.g., clay) becomes brittle under fast strain rate. To use moist gouge and clay as analogs of rocks, the appropriate strain rate should be determined under which they have similar response to strain as the rocks do under natural strain rate. An appropriate strain rate is determined by using the definition of viscosity. For simplicity it was assumed that moist gouge (gouge and water mixture) and clay (clay and water mixture) are Newtonian materials, which is approximately true when water content is high (Mitchell, 1993), then the strain rate is derived from the definition of viscosity £ = - (2.7) rj where a is the stress at a strain rate of e. The viscosity or rigidity of the crustal rocks (both granitic and mafic) varies in a large range, according to different studies. Long-term creep tests lasting more than twenty years at room temperature (Ito 1979, 1983) indicate that for very small strain rates of 10'12/s to 10"15/s the viscosity for granite is 1019 Pa-s and for gabbro, 1017 to 1020 Pa-s. From these data they inferred that viscosity of the oceanic crust is on the order of lO^-lO2 5 Pa-s, whereas that of the orogenic crust is on the order of 102 1 Pa-s. Geophysical studies suggest that the viscosity (rigidity) of the lithosphere is about 1023 Pa-s (Fowler, 1990), and the viscosity decreases with depth to 1019Pa-s (Yang & Toksoz, 1981). Taking the average between the viscosity giving by long-term tests (Ito 1979, 27 1983) and geophysical studies, the viscosity of the continental crust is estimated to be on the order of 102 1 Pa-s. For stress used to estimate strain rate, I choose critical stress or strength of materials, because materials have the same response to the critical stress- failure and fracturing. Uniaxial compressive strength of granite and diabase are 2.33xl08 Pa and 4.86xl08 Pa, respectively (Jaeger & Cook, 1976). Static rigidity (strength) of the crust is generally accepted to be around 101 0 Pa. Taking the average strength of crustal rock in the order of 108 Pa, the strain rate under the critical stress (strength) for the crust rock is then As comparisons, strain rate in ductile shear zones (Pfiffner & Ramsay, 1982; An & He, 1987) and average flow rate of the lithosphere (Carter & Tsenn, 1987) are in the order of 10'13/s Viscosity of clay and fault gouge with water content at about 39 wt.% to 41 wt.% were not directly available. Ramberg (1981) reported that at the stress level above the yield stress, modeling clay has viscosity in the order of 107 Pa- s. The modeling clay is a mixture of clay and oil with density of 1.71g/cm3. Since the yielding stress of the modeling clay is in the same order as the clay we used (Mitchell, 1993), we took the value 107pa-s as a rough estimation of viscosity for clay and fault gouge. Actual testing of the mechanical and rheological properties of fault gouge and clay with the exact amount of water needs to be done in the future for a better scaling relationship between rock and rock analogs we used. Strengths of our moist clay and gouge are estimated from Mitchell's data about clay containing about 39 wt.%. This estimation gives a value of about 103 -104 Pa (Mitchell, 1993). 28 The strain rate at failure for clay, taking strength as 103 pa, is therefore on the order of: • 103 m-4 - i £_ = —= - = 10 S . m 10 The ratio between the two strain rates is: r = ^ l = 10\ £c This means the strain rate of the experiment should be about 108 times faster than that of crustal deformation. Taking the maximum strain in the model as 1.5, then it would take 4.75x104 years for the natural rock in the crust to achieve this amount of strain, while it need only 4.16 hours for the experiment. All the experiments were finished by controlling the strain rate at about KH/s and the running time was about 4 hours. 2.6 Monitoring of the strain Both circular and square strain marks were inscribed on sample layers to monitor strain. Circular marks were useful in monitoring strain in all directions, and square marks were preferable for monitoring simple shear strain. Two methods were used to calculate strain. One was to measure the distorted original circular or square marks. For an original circular mark, strain can be derived by solving the equation (Ramsay and Huber, 1983): x 1 + 2ycy + (\ + y2)y2 = 1 (2.8) where x and y are the coordinates of any point on the elliptical marker and y is the shear strain. For a square mark, the shear displacement d on the vertical side and original side length of the square / were measured and strain 29 was calculated as d/l. The other method was to measure original slab width w and shear displacement d. The strain was d/w. 2.7 M onitoring of structure development The onset and evolution of structures within a sample layer are followed visually and photographically. Low-angle lightening was used from different directions to enhance the structures during photographing. More detailed analyses were performed later by studying these pictures. To avoid boundary effects, only the structures developed well within the boundaries were analyzed. An experiment was stopped when the sample layer lost coherence and when the board had been tilted to 90°. 30 Chapter 3 FAULT DEVELOPMENT IN FINE GRANULAR MATERIALS: FAULT GOUGE AND CLAY EXPERIMENTS 3.1 Introduction If a granular material (e.g., clay or fault gouge) consists of only fine particles (<125 pm) which are uniformly mixed with water, the material tends to be relatively cohesive. If, on the other hand, the particle sizes vary over a large range (the full range of particle size in the gouge samples is known to vary from <2 pm to >16,000 pm, see An & Sammis 1994) or preexisting structures (e.g., fractures, bubbles, inclusions) exist, then the material tends to be relatively incohesive. The experiments with relatively cohesive sample layers are presented here. The experiments with relatively incohesive layers will be presented in the next chapter. In a cohesive granular material, fault development is less affected by various preexisting structures (particle boundaries and fractures, for instance). The fault pattern developed in such a material can be used as a "model": it can be used to test theoretical predictions of fault development, it can be used to explain fault patterns developed in simple geological settings (simple lithologic unit such as a basalt or tuff layer, simple stratigraphy as in Quaternary basins, or simple tectonic history such as historically stable regions), and it can be used to compared to fault patterns developed in incohesive and heterogeneous materials and thus allows us to estimate the effect of incoherence and heterogeneity on fault development. It will be 31 argued in Chapter 4 that a cohesive granular material can be used to simulate fault development at depth in the earth's crust where confining pressure is high and materials are more ductile. This chapter focuses on the following ten characteristics of fault evolution: (1) nucleation, (2) propagation, (3) interaction and coalescence, (4) fault pattern, (5) geometry of an individual strike-slip fault, (6) evolution of fault steps and pull-apart basins, (7) fault rotation, (8) the scaling relationship between fracture length and propagation rate, (9) the scaling relationship between fault length and shear displacement and (10) the effect of material properties on fault development. The fault patterns produced in the experiments will also be compared with other experimental models and with natural fault patterns. 3.2 Sample preparation Clay and fine fault gouge from the San Andreas and the San Gabriel faults were used to make more cohesive experimental layers. Two types of fine gouge samples were prepared with different water contents and upper grain size cutoffs to test the effect of water content and grain size on fault development. The first type, subsequently referred to as "less moist gouge," was made of gouge material (grain size than 125 pm) mixed with 39wt.% of H 2O. The second type, subsequently referred to as "moister gouge,” was prepared by filtering out grains larger than 62.5 pm from the first type and then mixing the retained finer grains with 43wt.% of H 2O. Water content in the clay was also set at a level of about 39wt.%. The water allowed the material to transmit stress during deformation, and yet remain soft enough to 32 deform under the gravity loading. Each of the three types of materials had different mechanical properties. The two types of fault gouge were relatively "brittle" while clay was more cohesive and ductile. The moister fault gouge was less cohesive than the less moist gouge 3.3 Experimental observations Figs. 3.1 to 3.3 show fault development in the three types of granular materials during the simple shear experiments. Faulting sequences in the three types of materials are given in Table 1. It was found in the experiments that fault and fracture should be distinguished because a fault contained different types of fractures. Therefore in the following description, a fracture refers to a simple form of discontinuous structure which does not involve any coalescence, while a fault refers to a compound fracture which includes at least two coalesced fractures. A shear fracture is also called a shear. With these definitions, it was found that fault development in the experiments experienced three stages: protofracture stage (fracture nucleation), fracture stage (fracture propagation) and fault stage (fracture coalescence). A fault formed by fracture coalescence again coalesced with other faults forming an even larger fault. Both coalescence and displacement along strike-slip faults created various types of structures. These and other related characteristics of fault development are described in detail in the following. 33 Fig. 3.1 (continued) Fig. 3.1 (former two pages) Fracture and fault structures developed in a less moist gouge layer under simple shear. Fracture sets are identified in the inserts in the lower-left corner of each photo. At stage (a) the layer dilates. Also note the conjugate protofractures that are very faint at this stage. Some fractures nucleated at preexisting defects. At stage (b) primary shear Si and Si' sets develop from the protofractures and pores. The Si and Si' shears are equally developed at this stage. Proceeding to the stage (c), secondary shear set S2 emerges at a few locations. Si develop further by in-plane growth and out-of-plane coalescence. Note the development of a large shear zone near the center which carries the largest offset. The final stage is shown in (d), where a through-going shear zone has been developed. Note the tension component indicated by the pull-apart offset. 36 37 Fig. 3.2 (continued) Fig. 3.2 (former two pages) An experiment with moister fault gouge, (a) shows conjugate shear fractures Si and Si', and tensile fracture T. (b) shows Si and Si' shears cross each other, forming a fracture grid. Note that T fractures merge into compound fractures (faults), (c) demonstrates development of secondary shear set S21 as deformation continues. The layer finally tears apart along compound fractures, as shown in (d). Fracture sets are identified by the inserts in the lower- left. The length of the ruler is one foot. 39 Fig. 3.3 (continued) Fig. 3.3 (former two pages) Faulting of a clay layer under simple shear, (a) is the development of conjugate protofractures. Some of them have evolved into shear fractures Si and Si', (b) shows well-developed Si shears and much less developed Si' shears, (c) demonstrates that as strain increases, secondary shear set S2 appears which connects Si fractures in left-step. Note the dome (arrow) and rhombocasms formed by fault displacement. Finally, in (d), a through-going shear zone is developed. The zone is most recognizable from the largest offset it carries. Fracture sets are indicated by inserts in the lower-left. 42 Table 1 Faulting sequence under simple shear stages | strain structures (angles*) | coalescence remarks less moister gouge layer dilation and proto- ffacturing -0.15 open pores, protofractures, frac tures (21±2°/79±2°) conjugate protofractures & fractures fracturing -0.3 Si (22±2°) St'(80±2°) micro shears S i-S i’ conjugate coalescing -0.4 s 2(~i°) rhombocasms S i-S !1 Si-S2 Sj(21±2) S i'(83±3) faulting -0.5 TS zone (-16°), rhombochasm, T-fractures TS-Si, TS- Si', t s -s2 Si(22±2°) Si'(83±4°) moister gouge layer dilatation and proto- ffacturing -0.2 open pores, micro fractures, protofrac tures (19+2°/81±2°) conjugate protofractures & fractures shear fracturing -0.35 Si (19+5°) Si'(82±2°) micro shears S i-S j' conjugate Tensile fracturing -0.5 T (30°-60°) S i-S i' Si-T Si(19±5°) Si'(94±3°) coalescing -0.75 S2 (10±3°), S2'(76±2°), rhombochasms Si-Si', Si-T S 1-S2, S i-S i1 Si (23+7°) Si'(102±2°) faulting -1.2 pull-apart holes - Si(25±9°) Si'(103±3°) clay layer dilatation and proto- fracturing -0.15 open pores, protofractures (13±2°/82±4°) conjugate protofractures & fractures fracturing -0.3 Si (14±2°) Si’(84±4°) micro shears S i-S i’ conjugate coalescing -0.36 S2( -0°), rhombo chasms, upwarps S i-S i’, S1-S2 Si(15±2°) Si’(85±4°) faulting -0.39 TS zone (8°) TS-Si, TS- Si', TS-S2 Si(16±3°) Si'(86±4°) *the orientation of fractures with respect to simple shear direction. **TS: through-going shear zone 43 3.3.1 Protofracture stage The very beginning of the fault development was the development of low-displacement protofractures in defect-free areas (Figs. 3.1a and 3.3a). A sample first experienced dilation as evidenced by pore opening and dry-up of water on the sample surface due to absorption by the open pores. Then a special type of protofractures emerged in defect-free area. The protofractures were conjugate: one set trended 13°-22° and the other set trended 79° - 82° clockwise from simple shear direction (Table 1. The simple shear direction is indicated by half arrows in Figs. 3.1 through 3.3). The protofractures were faint but very dense: the intervals between the protofractures were only around 0.5 mm. The protofractures exhibited no measurable offset but their lengths were tens of centimeters as soon as they became visible. In the areas where defects (mostly visible pores) existed, fractures nucleated directly on the defects and began to propagate in conjugate shear directions (Figs. 3.1a and 3.3a). Only a small number of fractures nucleated in this way. 3.3.2 Fracture stage As deformation continued, shear displacement along some of the protofractures became detectable, and the protofractures then became shear fractures. The two conjugate fracture sets are termed primary shears. The synthetic set is represented by symbol Si and the antithetic set, by Si'. It was notable that only a small number of protofractures became shear fractures. The remaining ones between these fractures were disabled (the disabled protofractures can be seen in Figs. 3.1b and 3.3b) as all the shear deformation 44 was taken over by the shear displacement along the shear fractures. The intervals between the shear fractures were approximately equal in a granular material: about 12 mm in the less moist gouge layer, 5 mm in the moister gouge layer and 7 mm in the clay layer. Once nucleated, the shear fractures propagated in their own planes (Figs. 3.1b 3.2b and 3.3b). The Si shears propagated along a direction which varied between 13° and 22° in the three different materials measured clockwise from the applied simple shear, and Si' propagated along a direction from about 80° to 84° from the simple shear (Table 1). The propagation tips of the fractures were vague, first becaming one low displacement surface protofracture and then gradually fading away (Figs. 3.1b and 3.3c). At the later stages when fractures grew larger and began to interact to form compound fractures (strike-slip faults), the fracture and fault tips became more complicated (Figs. 3.3 and 3.4). Such fractures and faults generally terminated as en echelon shear fracture arrays, horsetail fractures, or splay fractures. The horsetail fractures were mostly shear fractures that are similar to the major fractures but some of them might have more tensile component because of slightly different orientations. The fractured regions around fault tips were similar to "breakdown zones" or "process zones" (Freidman et al.1972, Evans et al. 1977, Cox and Scholz 1988, Fig. 3.4 arrows). As numerous fractures grew in length they became closer each other. At the same time, one or two conjugate sets of secondary shears S2 and S2' and tensile structure T nucleated in the space between the primary shear fractures (Figs. 3.1b, 3.1c, 3.2c and 3.3c). Fractures then began to interact. 45 Fig. 3.4 Strike-slip faults terminated as protofractures, single shear fracture, en echelon shear arrays, or horse tail fractures, (a) shows faults terminated near the boundary of a clay layer, and (b) shows faults terminated in the interior of a clay layer. Also note the echelon micro fractures within the shear structures in (b). 46 3.3.3 Fault stage Strike-slip fauls began to develop once the fractures started to interact and coalesce. Fracture interaction and coalescence usually took place by taking advantage of existing structures. Occasionally interaction and coalescence occur over a structure-free area by creating a new bridge structure. The earliest interaction was between the conjugate primary shear sets Si and Si'. The two fracture sets crossed over and offset each other, creating segmented fracture traces (Figs. 3.1b and 3.2b). Later secondary conjugate shear sets S2 and S 2 ', as well as tensile fracture set T, were added to the fracture pattern (Figs. 3.1c, 3.2c and 3.3c). They were small fractures developed only between the parallelogram frames formed by the primary fracture sets and were terminated by the primary fractures when they met them. However these fractures were important bridge structures which linked-up two larger fractures during fracture coalescence. Fracture coalescence mostly happened between fractures within Si set and less often between the fractures within the conjugate Si' set. The Si shears were linked up through Si', S2 , S2 ' shears and T fractures forming a large compound fracture~a strike-slip fault (Figs. 3.1b and 3.3c). When coalescence occurred between two Si shears through an Si' or T fractures, a releasing step was formed (Figs. 3.5a and b, Fig. 3.6). When coalescence occurred between two Si shears through an S 2 shear, a restraining step was formed (Figs. 3.3c and 3.5c). Under most circumstances, more restraining steps were formed along a strike-slip fault than releasing steps (Figs. 3.1 and 3.3). A restraining step formed by S2 was really a shear structure at the beginning but acquired a compressional component during fault 47 (a) (b) (c) Fig. 3.5 Diagram showing coalescence of fractures, (a) and (b) show two Si fractures link up through a Si' and T fractures, respectively, making right-steps, while (c) shows two Si fractures link up through a S 2 shear making a left-step. displacement along the Si shears. No fracture was observed to link up with another fracture through a compressional structure. A similar coalescence process happened later between different faults. Such coalescence was a major mechanism of fault growth. Coalescence not only made faults grow very fast but also made strike-slip faulting process more chaotic. A short fault sometimes linked with a longer fault forming an even longer fault while a longer fault sometimes grew very slowly because no coalescence was possible. In a fault pattern where numerous strike-slip faults competed to grow by propagation and coalescence, one fault grew faster than the other faults, spanned the whole region and became a through-going 48 Fig. 3.6 Experiment with less moist gouge showing Si fractures are coalesced through Si' and T fractures forming compound fractures. Note the formation of rhombocasms by fault offsetting. Also note that protofractures between faults are disabled, and their trends are different from those of strike-slip faults. 49 shear zone (Figs. 3.2d and 3.3d). This zone was again not parallel to the applied simple shear but oriented around 10° (about 15° in less moist gouge and 8° in clay) from it. Deformation became noticeably heterogeneous upon the emergence of a through-going shear zone. Strain was predominantly accommodated by the through-going structures thereafter and soon the samples failed along the through-going structures. Such a through-going shear zone did not develop in the moister gouge layer. 3.3.4 Fault patterns Fault patterns developed during these simple shear experiments were fault grids consisting of several fault sets (Figs. 3.1 to 3.3). A fault pattern at its early stage contained only a few fracture sets, usually Si (primary synthetic shear) and Si' (primary antithetic shear, Figs. 3.1b and 3.3b). Both Si and Si' fractures were approximately regularly spaced. The spacing varied with materials (~7 mm in clay and ~12 mm in less moist gouge) and water content (the spacing was ~5 mm in the moister gouge with 42wt% H2O, Fig. 3.2b). Fracture traces were simple, straight and continuous. A through-going shear zone was not yet developed. In the experiment shown in Fig. 3.2b, a domain structure was developed where Si was predominantly developed in some areas whereas Si’ was predominantly developed in other areas. As deformation continued, more structures nucleated: first S2 (secondary synthetic shear) and T (tensile fracture), then S2' (secondary antithetic shear). The S2 was the only shear set which was about parallel to the applied simple shear. The T fractures mostly occurred in less cohesive gouge layers (Figs. 3.1 and 3.2), and S2 ' was only observed in the moister 50 gouge shown in Fig. 3.2 which had a high strain ( y=1.5). In the experiment with clay, a set of surface lamella also developed parallel to the maximum compression plane (Fig. 3.3c). All these structures were much shorter and had less displacement than the primary shear sets Si and Si'. They were also not regularly spaced but rather irregular and scattered. In the whole fracture pattern, Si was the most developed set in that these fractures had the longest length and largest displacement. The Si' set was less developed than Si although they emerged at the same time. After fracture interaction and coalescence, most of these structures disappeared. They were replaced by compound fractures— strike-slip faults, which had winding traces and contained fractures which originally belonged to different fracture sets. The most prominent strike-slip fault system in the pattern was the one about parallel to the Si shear. A through-going shear zone (sometimes more than one) emerged from this dominant set. The through-going shear zone was again not parallel to the applied simple shear, but was generally several degrees closer to the simple shear direction than the Si shears (Table 1). 3.3.5 Geometry of an individual shear structure An individual shear structure was either a shear fracture (a shear without coalescence) or a strike-slip fault (a compound shear which contained many fractures). A shear fracture was a straight and smooth linear structure at its early stage. As strain increased, numerous en echelon microfractures were found inside a macroscopic shear fracture (Fig. 3.4b). Under the right- lateral shear condition, all the echelon microfractures in an Si shear stepped left-laterally, and all the microfractures in an Si' shear stepped right-laterally. 51 The development of the microfractures changed the morphology of a macroshear fracture into feather-like structure (Fig. 3.4b). The microfractures were identified as shear fractures based on the observations that the microfractures had different orientations within Si and Si' sets and within each shear set the microfractures were oriented nearly parallel to the overall orientation of the macroscopic shear fractures in which they developed (the difference is <3°). A strike-slip fault generally had a winding trace which consisted of many fractures belonging to different fracture sets. The fractures stepped both right- and left-laterally in the same fault, forming fault steps. Under the right-lateral simple shear, a right-step created a releasing step where fault walls were pulled apart forming rhombocasms (Figs. 3.3c, 3.3d and 3.6), while a left-step in such an environment developed a restraining step where fault walls were pushed together (Figs. 3.3c and 3.3d). There were two types of releasing steps (Fig. 3.5). An S-type releasing step formed when two Si shears were linked up by an Si' (or equivalently two Si' shears were linked up through an Si). A T-type releasing step formed when two Si (or Si') shears were linked up by a T fracture. Under most circumstances, more restraining steps were formed than releasing steps in such a right-lateral shear field. Two types of restraining steps were also observed (Fig. 3.7). A low-angle restraining step was formed when two Si shears coalesced through an S2 shear (Figs. 3.3c and 3.3d, also Fig. 3.7a and b). A restraining step formed in this way was actually a shear-dominated structure, although upwarps along such structures indicated that they had a compressional component. A high- angle restraining step was formed by conjugate shearing (e.g., in Fig. 3.2 Si' 52 tW t I (a) (b) low-angle type (c) (d) high-angle type Fig. 3.7 Sketches showing two types of restraining steps observed in the experiments, (a) and (c) show the geometry before displacement and (b) and (d) show the geometry after displacement. In (d) an old high-angle restraining step is destroyed but a new one is created. shears were offset by Si shears forming such steps; also see Fig. 3.7c and d). Such a step was generally smaller in scale than the first type, but it constituted a major barrier to fault slip since it had a high angle to a fault trace, as will be discussed in the following. 3.3.6 Evolution of fault steps and pull-apart basins After shear displacement along a winding strike-slip fault, the two fault walls became unmatched (Figs. 3.Id and 3.3d). The walls were in close contact at the restraining steps, and were pulled apart along the releasing steps (Fig. 53 Si (a) (b) (c) S (d) (e) Fig. 3.8 Sketches showing the types and evolution of pull-apart basins observed in the experiments. In (a) a basin is formed by pulling-apart along an S-type releasing step, in (b) a basin is formed by pulling apart along a T-type releasing step, and in (c) a basin is formed by shear displacement along an adjacent restraining step. A compound basin is shown developing from (d) to (e) by coalescence of several simple basins. 54 3.8a and b). The walls were also pulled-apart along regular shear segments (e.g., Si) when shear displacement took place on a restraining step nearby (Figs. 3.Id, 3.3d and 3.8c). After a fair amount of shear displacement, the two walls of a strike-slip fault contacted at only a few points on the restraining steps (Figs. 3.Id, 3.8d and e). These points were critical for further fault displacement and acted as fault barriers. Two types of barriers were recognizable corresponding to the two types of restraining steps (Fig. 3.7). A low-angle barrier had synthetic shear displacement (Fig. 3.7a and b). The contact area for such a barrier reduced with progressive shear displacement. When the contact area had reduced small enough the barrier was smoothed out. New contacts were shifted to other locations. On a high-angle barrier shear displacement was almost perpendicular to the fault strike, and strike- parallel slip was not possible without breaking through the barrier (Fig. 3.7c and d). Once some of the old barriers were broken, new barriers were formed by conjugate shear displacement on another fault (Fig. 3.7d). Such a process repeated so that barriers were continuously destroyed and created. A pull-apart basin (rhombochasm) formed on a releasing step (type I) was bounded either by four conjugate shear structures (an S-type step, see Fig. 3.8a), or by two shears and two tensile structures (T-type releasing step, see Fig. 3.8b). A pull-apart basin formed near a restraining step (type II) was bounded all by shear structures (Fig. 3.8c). The geometry of pull-apart basins changed with fault displacement. The type I pull-apart basins grew parallel to the fault strike, while the type II pull-apart basins grew oblique to the fault strike. Generally type II basins were narrower but longer than type I. When confining pressure was added by tilting the loading board laterally, the type II basins were less developed. The two types of pull-apart basins always linked 55 up forming larger compound basins along a fault trace as fault displacement increased further (Fig. 3.8d and e). 3.3.7 Rotation A fault pattern was a dynamic feature. Fracture and fault orientations listed in Table 1 are the initial orientations (which might already include some rotation). Their orientation changed during experiments with progressive deformation. The Si and Si' rotated clockwise in the right-lateral shear field as the deformation proceeded. The Si' had the largest rotation. From the beginning to the end of the experiments, the Si' had rotated 3° clockwise in the less moist gouge layer, 22° in moister gouge layer, and 2° in clay layer, respectively (Table 1). The maximum rotation occurred in moister gouge due to the largest shear strain (7=1.5). Si was relatively stable. Its maximum rotations were 0°, 6°, and 2° for less moist gouge layer, moister gouge layer and clay layers, respectively. The rotation was an expected consequence of finite strain simple shear and can be observed in the distortion of the grid markers. Before fractures develop (a little earlier than those shown in Figs. 3.1a and 3.3a), the horizontal grid lines had already been rotated. Most rotations after the development of fracture sets were caused by mutual shearing of different shear sets and fracture coalescence. Systematic displacement along a shear set cut and offset its conjugate shear set, causing the rotation of the overall fracture traces. The mechanism was well demonstrated by the offset grid lines in Figs. 3.1 and 3.3. The rotation caused by this mutual shearing was clockwise for Si' set and anticlockwise for Si set under the right-lateral simple shear condition. For strike-slip faults, the 56 rotation was also caused by fracture and fault coalescence. When several Si shears coalesced through S2 shears forming a compound shear (strike-slip fault), the orientation of the strike-slip fault was closer to the applied simple shear (Figs. 3.Id and 3.5c) than individual Si fractures. When several Si shears linked up through Si' or T structures forming a strike-slip fault, the overall orientation of the fault was farther away from the applied simple shear (Figs. 3.5a, b and Fig. 3.6). The rotation produced a change in the mechanical properties of the fractures. The Si' shears showed the most dramatic change. As they rotated away from their original direction, they came under increased compression, and their growth gradually stopped. Some of the Si' appeared to heal because of the compression caused by large amount of rotation (Fig. 3.2c). The Si became more tensile as the result of the rotation. 3.3.8 Scaling relationships between fracture propagation rate and fracture length The relationship between fracture propagation rate and fracture length were measured by choosing a number of shear fractures which didn't coalesce with other fractures, and measuring the initial lengths of these fractures, Lj, on a photo taken at an earlier time of fracture evolution, and then measuring the lengths of the same fractures, L2, on a photo taken later when shear strain had increased by -0.2. When L2 is plotted as a function of Li (Fig. 3.9), the data can be fit by a straight line L2=1.76Li with a correlation coefficient R=0.94 (a perfect fitting yields R=l), indicating that during this time interval, L2/L 1 was about constant regardless of the initial fracture length (Lj). In another word, the length increment (AL) of a shear fracture was linearly proportional 57 14 12 L = 1 .75L 2 R =0.94 10 8 6 4 2 0 3 0 2 1 4 5 6 7 L (cm ) Fig. 3.9 Plot of initial length Li against final length L 2 for fractures developed in a clay layer. The linear fitting of the data implies linear relationship between fracture length and their propagation rate. R is the correlation coefficient. to the original fracture length (i.e., AL=CL where C is a constant related to strain). Since propagating velocity v=AL/At, where At was the time interval during which fracture length was measured, and At was a constant for all fractures in the same pattern, v was also linearly proportional to fracture length. The linear relationship between fracture propagation rate and fracture length hold only for fractures (shear structures not involving coalescence). At the later stages of these experiments, coalescence became a wide-spread 58 mechanism, and fractures coalesced forming faults. Fault length and propagating rate were no longer linearly related. 3.3.9 Scaling relationships between fault displacement and fault length The fault lengths and the maximum displacement along each independent fault on a deformed clay layer shown in Fig. 3.3 were measured. The maximum displacement on each fault was measured from the dislocated grid markers going through the middle portion of the faults. Fig. 3.10 shows the fault displacement as a function of fault length. The correlation coefficient is R=0.90. Fig. 3.10 suggests a linear relationship between fault displacement and length. The ratio of displacement over length is about 0.01, which agrees with the ratio reported by Cowie & Scholz (1992) based on their 10 oo u=0.01 L R = 0.90 8 6 4 2 0 0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0 8 0 0 L (m m ) Fig. 3.10 Fault displacement u is plotted as a function of length L. The linear fitting suggests that the maximum displacement of a fault is linearly related to its length. R is the correlation coefficient. 59 field data. It also confirms the theoretical results that the maximum displacement on a fault is linearly correlated with fault length (e.g., Pollard and Segall, 1987). 3.3.10 Comparison of fault development in the three types of granular materials Fault development within the three types of moist granular materials shears the following similarities: each fault pattern began with two conjugate shear sets followed by secondary shear set and sometimes, tensile structures; most strike-slip faults initiated as low displacement surface protofractures (a few nucleated on defects) and then became fractures; the fractures subsequently evolved into faults by coalescence; and these shear fractures propagated in mode II. The biggest difference was in the orientations of the shear sets. In the clay layer the angle between the simple shear and Si shear set was the smallest but the angle between the simple shear and the Si' set was the biggest among the three types of samples (Table 1). The less moist gouge had the largest angle between simple shear and Si and the smallest angle between simple shear and Si'. The second difference was in the development of fracture sets. Both Si' shears and T fractures were more developed in the two types of gouge layers (they had larger populations, longer fault length and larger displacement, compare Figs. 3.2 and 3.3, for example). However the S2 shears were more developed in the clay layer. In the clay layer, a majority of Si shears were linked up through S2 shears. The compound shears (strike- slip faults) formed in such a way were closer to the applied simple shear direction than Si and had more shear component (Fig. 3.3d). In the less moist 60 gouge layer, Si shears were frequently linked up through Si' and T structures forming strike-slip faults. The strike-slip faults formed in such a way deviated farther from the applied simple shear than Si shear (Fig. 3.6) and had a larger opening component. In the gouge layer with higher water content, tensile faulting became very active at a later stage and no through- going shear zone was developed. The third difference was in fault density. In clay and moister gouge layers faults were denser (average interval between major faults was about 5 mm in moister gouge and 7 mm in clay) than in the less moist gouge layer where the average interval was about 12 mm. Also notable was a set of dense but faint compressional surface lamella oriented approximately 60° anticlockwise from the simple shear in the clay (Fig. 3.3c). The lamella was not obvious in the two types of gouge layers. 3.4 Discussions Many of the observations, i.e., fracture nucleation, propagation, origin of en echelon fault geometry and grain size effect on fault development, will be studied in detail in the following chapters (Chapters 4 through 6). This section only discusses the orientation of major shear fractures and strike-slip faults, and compares the fault patterns from the experiments with Riedel model and with natural fault patterns. 3.4.1 Implications of protofractures The development of protofractures before fracturing may indicate that once a material (not necessarily granular) is subject to a simple shear stress, all 61 the defects in the material can be activated simultaneously. The defects first dilate which leads to strain hardening, then may experience certain shear growth, and finally coalesce with each other forming long protofractures. The high density of the protofractures in the moist clay and gouge may indicate large population of the active pore defects. The long length of the protofractures may indicate strain hardening which makes the material brittle. The conjugate geometry of the protofractures indicates that they are shear structures. Most of the protofractures are disabled after some grow into fractures. This may suggest stress relaxation caused by shear displacement along the fractures. The spacing between fractures seems controlled by layer thickness, water content and material types. Initial observations indicate that a thicker layer with lower water content has larger spacing between fractures. It is still questionable if these structures can be observed in the crust. It is probable that protofractures can not develop in near surface hard rocks, but they might be seen in the rocks coming from the deep depth as well as in soft sedimentary layers in basins. Rocks in the deep depth are subject to large confining pressure and are more ductile. These conditions favor shear fracture development (Chapter 6). Some weak surfaces which eventually evolve into straight, equally spaced joints after uplifting may be related to protofractures formed earlier in the deep depth. Some early forliations in metamorphic rocks may also share similarities with protofractures. Unsolidified sediments in a sedimentary basin are the closest analogue to the granular samples used in the simple shear experiments. However small scale protofractures can be easily erased or disturbed by topographic relieves, 62 although regularly spaced lineaments are usually recognizable on satellite images. 3.4.2 Why major shear fractures and strike-slip faults do not parallel to simple shear direction? The first unusual phenomenon noticed during the experiments was that no major shear fracture or strike-slip fault was parallel to the simple shear direction. The most developed shear set, Si, had an angle ~14°-22° from the simple shear in the three different materials. The through-going shear zones in clay and gouge had angles 8° and 16° from the simple shear, respectively. By comparing with the results from axial-loading rock mechanics experiments (Jaeger & Cook 1976), it is proposed that the departure was caused by the internal friction of clay and gouge layers. The different angles observed in clay layer and fault gouge layers were related to the different friction properties of these materials, which may be related to different particle sizes and minerals. The layers with coarse grains (gouge) should have the largest departure angle. The quantitative departure from simple shear direction, a, can be estimated from the coefficient of friction, p. According to Jaeger and Cook (1976), the friction angle < p of a material equals 0=tan-J j U (3.1) A shear plane would occur in the direction from the principal stress a ,: 6 = - - $ - (3.2) 4 2 where the second term represent the departure angle from applied simple shear, i. e., 63 a=^(p = jtavijii. (3 .3 ) For moderately rough surfaces of typical rocks, p is between 0.51-0.75 (Jaeger and Cook 1976). This gives a=12°-18°. The friction coefficients of clay and gouge layers containing 39wt% to 43wt% water are not known. The experiments indicate, however, that they are almost the same as that of rocks because the a is 14°-22° (this may include some rotation). Experiments conducted by Sims (1993) for a wet clay with 50wt% H 2O indicates a friction angle of 28°, corresponding to a=14°. It seems that internal friction of clay is not so sensitive to water content. The relationship between particle size and friction angle will be discussed in the next chapter after more experimental observations were made. 3.4.3 Comparison with Riedel shear studies The Riedel shear experiment (Cloos 1928, Riedel 1929) is one of the earliest and most traditional method to study shear structures in the laboratory. It is still widely adopted in recent studies (Cloos 1955, Tchalenko 1970, Wilcox et al. 1973, Naylor et al. 1986, Richard and Krantz 1991, Richard et al. 1991, Smith & Durney 1992, Schreurs 1994). Typical structures developed during a Riedel shear experiment are shown in Fig. 1.2 in Chapter 1. Typical structures developed in the simple shear experiments described in this thesis are assembled in the same way in Fig. 11. By comparison, it can be found that the primary shear Si and the conjugate primary shear Si' in the simple shear experiments correspond to the Riedel shear R and the conjugate Riedel shear R', respectively. However, no through-going shear zones (PDZ in Fig. 1.2) parallel to the simple shear 64 Sim ple. Shear (a) Fig. 3.11 Fracture assemblages developed under simple shear condition in this experimental study. direction were observed in the experiments. Also X, P and P' structures have not been observed in the simple shear experiments. One set of shear fractures was oriented in the X direction later during fracture evolution (Fig.3.2c). However, it was identified as rotated Si' rather than a new fracture set. By comparison, the secondary shear fracture S2 in these experiments was observed to lie parallel to the orientation of PDZ in Riedel experiment (Fig. 1.2), and the secondary conjugate shear S2 ' had an orientation very close to that of R'. They are both in the quadrant occupied by R and R'. No shear fracture was observed within the quadrant where P is supposedly to develop in Riedel experiments although the Si' in the simple shear experiments sometimes rotated a little toward that direction. Also, tensile fractures 65 sometimes developed in a direction close to the secondary conjugate shear P' in Riedel shear experiment. All the trends of the structures observed in the simple shear experiments were not stable but evolved with increased shear strain. For example, as Sp rotated away from its initial position and could no longer accommodate strain efficiently, S2' nucleated close to the original Si' direction (Fig. 3.2c). It has been observed that as Si' rotated approaching the orientation of X in Fig. 1.2 T structures were in the P' position. It thus seems likely that X and P' in Riedel experiments are not new, but rotated Si' and T respectively in the simple shear experiments. The differences between these experimental results and those from the Riedel experiments are probably due to the different boundary conditions. In Riedel shear a preexisting 'fault' between the two block boundaries is made underneath the clay layer. All the deformation in the clay layer is caused by the movement on this preexisting 'fault.' Therefore the structures developed in the top layer are secondary regarding the preexisting 'fault.' The through- going shear zone is not naturally developed but is forced to break in this way. Fracture interaction and coalescence are not demonstrated. Thus the Riedel experiment may only be suitable for studying surface ruptures caused by buried faults. My experiments did not have a preexisting fault as a boundary condition. All the fractures and faults were generated naturally under simple shear. Fault evolution process from low displacement surface protofractures through fractures to faults was all exhibited. Therefore my model might be more appropriate for studying the evolution of strike-slip faulting in broad crustal shear zones deforming in simple shear.. 66 3.4.4 Comparison w ith natural fa u lt patterns The fault patterns developed in the above experiments are morphologically similar to those found in Southern California and on Venus. Fault pattern in Southern California The late Cenozoic deformation in southern California has been simple shear related to the San Andreas faulting (Sylvester 1988). Two sets of strike- slip faults are dominant in this region (Fig. 3.12). One set is the San Andreas fault system which strikes northwest and has a right-lateral slip sense; the other is Garlock fault system trending almost east-west and having a left- lateral slip sense. The angle between the two sets of faults is between 110°- 130°. In the Mojave Desert, the two sets of faults developed in different domains which are postulated from paleomagnetic data to have rotated clockwise across a broad zone of simple shear (Luyendyk et al. 1985, Nur et al. 1989, Dokka & Travis 1990). It is proposed here that the early versions of the two strike-slip fault sets in southern California were shear fractures compatible to the primary shear Si and conjugate primary shear Si' in my experiments. At that early time the shear fractures in the region were relatively short and discrete but had a large population. The angle between the two shear sets was much smaller than they are now. Then interaction and coalescence turned some of the fractures into strike-slip faults. Numerous small faults competed to grow in this region during the process. At last one fault approximately parallel to Si set dominated to became a through-going shear zone— the San Andreas fault. The other less successful competitors became San Jacinto, Elsinore, Newport-Inglewood, and numerous smaller parallel faults in Mojave Desert. 67 OS 00 TEHACHAPI BLOCK \ * MORRO GABRIEL BLOCK BLOCK \ M in. F ault m ^ V CATALIN BLOCK 1 +'*36° 1 1 6 \ C E N T R A L \ MOJAVE X B L O C K ? X • > + \ V o u W \ \ EAST TRANSVERSE BLOCK ? WESTERN T R A N S V E R S E ^ F BLOCK 33°+ 1 2 1 ° 50 MILES 100 A Fig. 3.12 Fault pattern in southern California showing two sets of strike-slip faults in conjugate. Hatch pattern indicate domains where east-west set dominates (courtesy A. G. Sylvester 1988). In the conjugate fault set about parallel to Si' set there was also one fault which grew by coalescence to become the largest strike-slip fault in that direction— the Garlock fault. The other competitors become Pinto Mountain, Blue Cut, Malibu Coast, Santa Ynez, and similar faults in Mojave Desert. The fault orientations have been changed since due to further fault coalescence and finite shear strain. The Garlock fault system might have the largest clockwise rotation because it has had high angles with respect to the regional simple shear (estimated to be in the north-western direction). Assuming a fault is a straight line within a simple shear field, the rotation angle (o (respecting to the simple shear direction) of the line can be expressed. tanco = tanco'+y (y.4) which is proportional to the original angle of the line (o' and shear strain y. The increment of rotation to the original angle (o ' is: A(o = 0) - tan'1 (tanco- y). (3.5) Since the shear strain in southern California is large regarding about 240 km of post-earliest Miocene displacement along the San Andreas fault (Crowell 1962), the angle between the two fault sets has increased to 110°-130°. This means at least 20-40° of rotation was involved between the Garlock and the San Andreas fault systems. This also suggests that the two fault sets in southern California formed under simple shear condition. Fracture pattern on the Venus The Magellan radar images of Venus revealed striking linear features on its volcanic plains. The best known example is in the "gridded plains" of Guinevere Planitia where two sets of linear features can be identified (Fig. 3.13): One is faint and regular which form the NE trending component of the grid; another is brighter and less regular, forming the NW trending 69 component. The average spacing for the NE set is about 1 km and for the NW set, 2.5 km. Banerdt and Sammis (1992) identified the NE set as tensile fractures and NW set, tensile fracture with shear component. The NW- trending features are directly comparable to the strike-slip faults in my experiments, especially Fig. 3.4b, except the shear senses are different: both of them are relatively regularly spaced, having feather structure, and showing evidences of coalescence. The shear sense on the Venus might be right-lateral which can be inferred from the echelon arrangement of the shear segments within the shear structures. The conjugate shear fractures, however, are missing in this region, if considering the NE set as tensile ones. One explanation is that the shear strain in the NE direction has been accommodated by the preexisting tensile fractures, therefore mechanically it is not necessary to have the conjugate shear fractures. It should be pointed out that the comparison with natural fault patterns may be a complicated process and exact match may not be possible. The experiments performed here are for one episode of simple shear, whereas most natural fault patterns are developed through several episodes of deformation. Preexisting faults are not assumed in these experiments, while in nature older sets of faults generally exist before a new fault set is added to a pattern. The appearance of a natural fault pattern is generally affected by topography and the degree of exposure. Modeling these effects is possible in future with this gravity sliding technique. 70 Fig. 3.13 A Magellan radar image of the girded plain in Guinevere Planitia. Two sets of protofractures are developed on the plain: one trends NW and is relatively brighter, more irregular with identifiable en echelon segments; the other trends NE, and the fractures are regularly spaced and simple. This image is about 38 km across. JPL P-36699 MGN16. 71 3.5 Summary The simple shear experiments with moist clay and fault gouge layers demonstrated that: 1. Strike-slip faults were developed by the coalescence of shear fractures and shear fractures were nucleated as low displacement protofractures. The competent growth of the protofractures turned some of them into fractures while it disabled most of the others. The protofractures were conjugate upon nucleation. Some fractures nucleated on pores. 2. Once nucleated, shear fractures propagated in plane as shears. A fracture tip gradually became complicated by echelon and horsetail fractures as it grew longer. A majority of the fractures around the fracture tips were shear fractures. 3. Fracture coalescence led to the development of strike-slip faults. Coalescence occurred most often by taking advantages of existing fractures in the same fracture (fault) pattern. Both releasing- and restraining-steps were developed during coalescence. 4. A typical fault pattern formed under the simple shear condition was a fault grid constituted of several generations of conjugate strike-slip faults as well as tensile ones. The primary generation of conjugate strike-slip faults established the framework within which later generations were added. A through-going shear zone emerged from the most developed synthetic fault set. 5. Strike-slip faults did not develop parallel to the direction of regional simple shear. A through-going shear zone had a deviation angle about 10° from the applied simple shear. The deviation was suggested to be caused by internal friction of the materials. 72 6. A shear fracture contained echelon microshears stepping in one way, and a strike-slip contained echelon fractures and fault segments stepped in both ways. 7. Displacement along a strike-slip fault caused mismatch between the two fault walls which left a few contact points as resistant barriers and turned most of the other parts into pull-apart basins. Both releasing and low-angle restraining steps led to the formation of pull-apart basins. 8. Both fractures and faults rotated with progressive simple shear strain. 9. Fracture propagation rate was a linear function of fracture length, and fault displacement was a linear function of fault length. 10. The differences of fault development in the clay and gouge layers were embodied in structure sets and orientation: in the clay layer major strike-slip faults were closer to the simple shear direction. Antithetic primary shears were fewer and weaker but synthetic secondary shears were more abundant than in gouge layers. Tensile structures were more often developed in the gouge layer. Morphological similarities between the experimental and the crustal fault patterns suggests the applicability of the experimental observations to the study of crustal shear zones. 73 Chapter 4 FAULT DEVELOPMENT IN COARSE GRANULAR MATERIALS: EFFECT OF MATERIAL COHESION 4.1 Introduction Chapter 3 described fault development in moist granular materials-- clay and fine fault gouge. These materials were cohesive because of fine grain sizes. Structures developed in these materials were predominantly shear. Tensile structures were minor and mostly functioned as bridging structures to link-up two shear structures. A through-going shear zone finally developed along a direction about 10° from applied simple shear. In this chapter, a less cohesive material, coarse fault gouge, was used to perform the same experiment. The purpose of this study is to find the effect of material cohesion on fault development, including fracture and fault types, their propagation, orientation, and overall fault patterns. One of the important purpose of this study is to establish a relationship between grain size of a granular material and material cohesion, and further relate such a material cohesion to confining pressure in the crust. With such a relationship, it is possible to simulate fault development in the deeper crust (where confining pressure is high and material is more ductile) with cohesive gouge and clay, and to simulate fault development in the shallower crust (where confining pressure is low and materials are more brittle) with coarse and thus less cohesive fault gouge. In the following, the experimental observations of fault development in a coarse gouge sample are first 74 described, then the relationship between grain size and cohesion is explained. The similarity in the effect of capillary pressure in a granular material to the effect of confining pressure in the crust on fault development, and the dependence of friction angle on particle size in a granular material are also discussed. 4.2. Experimental Observations This section describes a simple shear experiment with coarse granular material — raw fault gouge, and compares the fault development process with those in fine granular materials described in Chapter 3. 4.2.1 Sample preparation Raw fault gouge from the San Andreas fault was used. The grain size within the gouge has been measured by An and Sammis (1994) and was found to vary from less than 1 pm to 16 mm. The raw sample was first disaggregated in water, and then dried after the disaggregation was completed. The dried sample was mixed with ~38wt% H 2O. The sample was then loaded onto the experimental apparatus following the procedure described in Chapter 2. In order to minimize boundary effects, the sample layer was made quite large with an overall dimension of 2.3x40x80 cm. Circular markers were inscribed on the sample surface before experiment to monitor strain. Since the samples were much softer than real geological materials, a scaled deformation rate of 10'4/s was used for the results of the experiments to be 75 applicable to real geological materials (Chapter 2). An experiment stopped when the sample failed. 4.2.2 Fault development in the coarse gouge layer Under simple shear, the coarse gouge layer first developed two sets of conjugate protofractures when strain reached 0.2, and then some of the protofractures evolved into synthetic primary shear set Si and the conjugate antithetic primary shear Si' (Fig. 4.1a). Both Si and Si' were zones of detectable width (~0.3 mm on average). Si shears were oriented 22+7° from the simple shear direction, and were dense and faint. Si' shears were oriented about 88±2° from the simple shear direction. They were also dense, but were less developed. Tensile fractures also nucleated at pores as well as at the boundaries of larger grains. Their orientation varied widely between 30° to 60° from the applied simple shear. The Si and Si' shear sets continued growing along the shear direction. Concurrent with the formation of shear fractures, more tensile fractures were nucleated throughout the sample (Fig. 4.1b). As these fractures grew and interacted with other fractures, the sample began to fail (tear apart) along some rapidly developing tensile fractures or faults (Fig. 4.1c). Note that even after the sample began to fail, displacement along each individual shears was very limited (Figs. 4.1c and d). Also note that the total strain reached by the sample is only 0.45. 76 Fig. 4.1 (continued) Fig. 4.1 Development of faults in a raw fault gouge layer under simple shear, a) shows the nucleation of primary shear Si and its conjugate Si'. A few tensile structures T also developed at this stage; b) and c) show more tensile structures developed in the following stages and d) is the stage when the sample failed along some tensile faults at large strain. The ruler in the figure is one foot long. 4.2.3 Comparison with fault development in cohesive materials Comparing with fault development in cohesive clay and gouge layers reported in Chapter 3, it is found that in both granular materials, conjugate shears Si and Si' developed when the layers were subjected to simple shear. They also shared the similarities that shear structures interacting in tension (i.e., right-stepping in right lateral shear or left-stepping in left lateral shear) generally link up by using old structures as well as developing new ones. However, many differences were also found. Displacement mode: There was a displacement mode change as the granular materials become less cohesive. In the more cohesive gouge and clay, shear was a dominant mode while tension was minor. The tensile structures, if observed, were mostly related to large pores created during sample preparation. In the less cohesive gouge, however, shear structures developed only at the very early stage. These structures accommodated less strain as compared to those in the more cohesive material (as evidenced by very small shear displacement). After such a short time, tensile faulting took place and became a dominant mode. It seems that the displacement mode 79 changes as particle size becomes coarser and as the granular materials become less cohesive. Friction angle: In the more cohesive gouge sample, the friction angle (the angle between primary shear Si and the applied simple shear) was about 13°-19° while in the less cohesive gouge, the friction angle increased to 22±2°. The only apparent difference between the two materials was in the particle size: the cohesive gouge had particle size <125 pm while the incohesive gouge had particle size as large as 16 mm. Therefore it seems that the large particle size increases friction angle. Theoretical analysis in the next section will justify this explanation. Fault width: a shear fracture in the less cohesive gouge layer was a break down zone of -0.3 mm in width at the early stage, while a shear fracture in a more cohesive layer was a very narrow and smooth discontinuity. Through-going shear zones: Through-going shear zones did not develop in the incohesive gouge, in contrast to cohesive gouge. The less cohesive gouge samples failed along tensile fractures. Strain: While the more cohesive samples achieved strain as large as 1.5 before failure, the less cohesive samples failed at lower strain (around 0.45 for the sample used). The less cohesive gouge broke down earlier than the more cohesive gouge. Strain distribution: In the cohesive samples, strain was distributed almost uniformly across the sample, except close to the boundaries. However in the incohesive samples, strain concentrated in mechanically weak areas, such as large pores, and particle boundaries. 80 4.3. Analysis The experiments above demonstrated that fault types and friction angle in fine and coarse granular materials are different. Because grain size is related to cohesiveness of a granular material, fault type and friction angle are also related to material cohesion. In this section, a theoretical relationship is established first between particle size and cohesiveness of a granular material. Then the relationship will be extended to relate material cohesion to confining pressure. These results will be used to explain fault displacement mode change with material cohesion. The increase of friction angle with particle size will also be explored in the section. 4.3.1 Relationship between particle size and cohesion In the previous sections, some granular materials were identified as more cohesive while others as less cohesive. The apparent criterion for this classification is the particle size: if particles are large, the material appears to be 'less cohesive'; if all particles are small, the material appears to be 'more cohesive.' This statement is qualitative and not scientifically robust. How small should the particles be to be considered as cohesive? Since it has been observed in the simple shear experiment that shear (mode II) fractures develop in more cohesive granular materials while tensile (mode I) fractures develop in less cohesive layer, a more interesting question is raised: at what threshold in terms of particle size, does the displacement mode switch? In this section adsorption and capillary pressure will be used to explain the relationship between material cohesion and particle size. Adsorption is 81 caused by free surface energy. Adsorption force in a unit volume of granular material is proportional to the specific surface area (total free surface area in a unit volume). Assuming the particles are spheres, then the specific surface area A* is < « > where A is the total surface area of the spheres in a volume V and d is the diameters of the spheres. Eqn. (4.1) indicates the increase of A* with particle size decreasing. Thus adsorption force is expected to be stronger in finer granular materials than in the coarser ones, and the finer materials should be more cohesive. Capillary pressure applies to moist granular materials. If the particles are water wettable, the magnitude of capillary pressure indicates how strong the water drops are held in the capillaries between particles. If capillary pressure is high, water drops will be strongly held between the particles and the granular material will be more cohesive. If the capillary pressure is low, then water drops will tend to be move free between particles and the granular material will be less cohesive. The sizes of capillaries in a granular material are related to particle sizes. The packings of the particles create pores and the pores connect each other forming capillaries. For cubic packing of spheres of equal size, the porosity 0 can be calculated to be: The volume of a pore can be expressed as Vp =ad3 where d is the diameter of the spheres. Approximating the pores as spherical in shape, then the pore radius, rP i is: 0=1-^/6=0.4764. (4.2) = 0.4845d. (4.3) 82 Similarly for rhombohedral packing of equal spheres, the calculated porosity is: 0=1'3 W =0'2595 (4'4) and the approximated spherical pore radius is: rp =0.3525d. (4.5) For simplicity, it is assumed that every capillary is a tube of radius rp. The capillary pressure, Pc , is then related to the radius of a capillary rp, and sphere size, d, by P . 2yvcosg _ 2ytcosfl rp Cd where yt is the surface tension of a fluid, d is the contact angle between a fluid surface and a solid, and C is a constant related to the sphere packing type. Eqn. (4.6) indicates an inverse relationship between capillary pressure and particle size. For water, ys takes 72.75 dynes/cm, 6 equals to zero for water/quartz interface, and C takes value of 0.4845 for cubic packing and 0.3525 for rhombohedral packing as discussed above (also see Dullien, 1992). Fig. 4.2 shows the increase of capillary pressure Pc with the reduction of particle size d. The solid line represents the case where particles are in cubic packing and the dashed line represents the situation where particles are in rhombohedral packing. Fig. 4.2 indicates that when d>100 |im, capillary pressure is negligible, and a granular material consisting of such particles is not so cohesive. In the above estimation of capillary pressure the simplest conditions have been assumed. In reality a capillary does not keep the same size along its length (Fig. 4.3a). Pores constitute the wider portions of a capillary and 83 xlO6 capillary pressure rhombohetral a. a. cubic 10-1 10° particle size d (microns) Fig. 4.2 Capillary pressure Pc is plotted as a function of particle size d. assuming partially water-saturated quartz particles. these pores are connected by narrow throats (Fig. 4.3b). If all pores and throats are occupied by water, no capillary pressure exists. If the capillary is only partiallysaturated by water, the water will choose to stay in the throats where capillary pressure is higher. The stagnated water drops in the throats will isolate the pores. When such a granular material is subject to a tensile stress, pore pressure will drop due to dilation. If the throats are large enough, the 84 capillary pressure is low and the outside pressure can drive the water to flow inward to release the pore pressure drop. Because the pore pressure can be kept at a constant level, the pores can continue to expand as fractures, and the Fig. 4.3 A capillary in a granular material in (a) consists of pores and throats as shown in (b) material appears to be not cohesive. If the throats are very small and thus capillary pressure is very high, the outside pressure can not drive the water drops off the throats. The low pore pressure will resist further dilation and potential tensile fracturing. The material therefore appears to be cohesive. Another complexity of granular materials used in the simple shear experiments is that those materials have a range of particle size distributions. For examples, the gouge sample shown in Fig. 3.1 has a maximum particle size of 125 p . but more than 80wt.% of the particles are smaller than 62.5 p in sizes. Because the small particles tend to fill the pores formed the large particles, the material still has small capillaries and it appears to be cohesive. 4.3.2 Displacement mode switch with grain size OOOO 0 3 0 0 throat pore (a) (b) 85 The switch of fault displacement mode with grain size observed in the experiments can thus be explained based on the above study. The fine granular materials (clay and fine gouge) were cohesive and capillary pressure inside was high. Pores in the material were sealed by water drops held in the capillary throats. During sample dilation which appeared at the beginning of every experiment, the pores were expanded and pressure inside dropped. Because of the high capillary pressure, the low pore pressure was not able to be released by water or air inflow. A pressure difference inside and outside the sample was thus developed. This corresponded to a condition of high confining pressure. Melin (1986) has concluded based on theoretical analysis that under high confining pressure mode II propagation takes place. Chapter 6 of this dissertation will elaborate an idea suggesting that under a low ratio of fault internal pressure versus confining pressure, mode II propagation takes place preferably over mode I. A confining pressure in excess of the pore pressure in a fine gouge and clay layer satisfies this condition. In the coarse gouge layer, on the other hand, capillary pressure was low according to Fig. 4.2. This corresponds to a material subjected to a low confining pressure. Because confining pressure and pore pressure were always balanced (details in Chapter 5), mode I propagation took place instead. 4.3.3 Friction angle increase with particle size As observed in the experiments, the friction angle a was larger in the less cohesive (coarse-grained) layer than that in the more cohesive (fine grained) layers. This means that the internal friction coefficient p. may be larger in less the cohesive layer (because fi-ta n la ). 86 bulk shear local shear (a) (b) Fig. 4.4 a) illustrates simplified asperities with flat-top saw teeth geometry, b) shows the geometry used to calculate forces. An apparent explanation is that large particle sizes increase the roughness of a shear plane, and thus increases friction. When a sliding surface is rough, sliding becomes uneven. At the stage when two asperities (or two particles) are in contact at their peaks, frictional force is little; after the two asperities slip into an interlocking position, the frictional force can be very large. Considering a simple situation where asperities are a series of flat- top saw teeth of similar geometry in 2-D (Fig. 4.4a). When two asperities contact on their flat top, the friction force F can be described by the conventional expression F=liN (4.7) 87 roughness and mu mu=0.6 -20 -40 -60 0 60 20 40 80 100 120 140 160 180 f.3 (degrees) Fig. 4.5 Variation of F/N (normalized force needed for slide to occur) with where /J . is the friction coefficient, and N is the normal force. On such a flat surface, sliding occurs when the applied shear force is equal to or larger than F. When the two asperities slip into interlocking position, however, the resistant force increases. For sliding to occur in this position, the force balance requires Fcos/3 - N sinfi = fi(Ncosf3 + Fsin(3) (4.8) where f3 is the angle between the bulk sliding surface and the profile of the asperity, and N is the normal force applied on the bulk shear plane (Fig. 4.4b). 88 The magnitude of the frictional force (or the bulk shear force needed to overcome the friction to make sliding happen) is then: p,_ N(sinp + ilcosp) c o sp - jlsinfi ' For 0°<f5<cotAn°, F increases with p until it reaches + < » (Fig. 4.5). At P=cot' 1 H, F is singular. The singularity represents a stagnant point at which neither forward nor backward slips are possible (F =±°°). For cot'lin<P <180°, F increases again with (5 but from until it returns to the value for a flat surface F=/J.N . The /J . in (4.8) refers to a frictional coefficient of a flat surface where sliding is exactly parallel to the sliding surface. For rough surfaces, the bulk sliding direction may not be parallel to local sliding direction because of asperities, as shown in Fig. 4.4b. Comparing (4.9) with (4.7), it is found that the bulk friction coefficient /U b for a rough surface can be written as F sin/J + jUcos/J ~ N ~ c o S j6 -/j» in ff • (4 1 0 ) The roughness of a sliding surface is reflected in P, an angle between the bulk and local constrained sliding directions (Fig. 4.3b). Because it is assumed that all the asperities are similar in geometry, p represents the slope angle of each asperity. Eqn. (4.10) indicates that the bulk friction coefficient is only affected by slope angle of asperities, not their size (asperity height). However for slip to happen on a rough surface where asperities are not destroyable, lateral tensile displacement (opening) must happen to ride over the asperities. The amount of lateral displacement required is the height of an asperity h . If a fault is sealed from outside, the pressure within the fault will drop as opening displacement occurs to ride over the asperities. A sealed condition is possible in the experiments as well as in the crust because of 89 water (Chapter 5). If water is not percolated, then it simply works as a seal agent. If water is percolated, then it still need time to flow into a suddenly opened fault gap which gives a transient sealed condition. According to gas law PoVo/To=PV/T, where Po, Vo, To and P, V, T are initial and changed pressure, volume and temperature, respectively, and assuming constant temperature within the fault before and after the opening displacement, the pressure drop can be found in relation to volume increase or asperity height through: = = (4.11) where ho is initial opening between two asperities, and h is final lateral offsets or asperity height. For two faults of similar asperity geometry but different asperity height h % and hi, the ratio of the pressure within the faults as lateral displacement reaches hi and hi is simply the inverse of the asperity height ratio A = (4.1.2) P2 Vx \ where Pi, Vi and Pi, Vi denote pressure within the faults and volume of the fault openings with asperity height hi and hi, respectively. Eqns. (4.11) and (4.12) indicate that large asperity height causes a large pressure drop. Now the bulk normal force N b should be the summation of N and internal pressure drop AP=P-Po if the fault is sealed: N b=N-AAP, (4.13) where A is the area the pressure drop AP is applied on. Eqn. (4.9) should then be rewritten to include the effect of pressure drop. Using Eqns., (4.11) and (4.13), Eqn. (4.9) becomes F = [N — AP0& -1 + (4.14) h cosp-/tsinp 90 asperity hight and mu 0.96 mu=0.6, beta=10; a/n=10^-7 0.95 0.94 0.93 0.92 £ 0.91 0.9 0.89 0.88 0.87 0.86 0.3 0.4 0.1 0.2 0.5 0.6 0.7 0.8 0.9 h (mm) Fig. 4.6 Plot of bulk friction coefficient fib versus asperity height h. Eqn. (4.14) directly relates resistant force F to asperity height h. Comparing Eqn. (4.14) with Eqn. (4.7), the bulk friction coefficient is (415> N h cos/J-/zsin/J Fig. 4.6 shows a plot of fib against h, assuming N /A - 107 Pa, Po=l06 Pa, n=0.6, ko=0.001 mm and /)=10°. It can be seen that fib increases quickly with asperity 91 height h at the beginning, then stabilizes at about 0.955. The /ib derived in this way is large because when slip occurs along downward slope of the asperities (i.e., /? is negative), the slip can occur much easier than along the upward slope according to Fig. 4.5, therefore jib is small. The corresponding friction angle can be derived by using: o c = ^tan~' fib. (4.16) This is the angle shear faults develop away from simple shear direction. It thus increases with sliding surface roughness as shown in Fig. 4.6. Since surface roughness increases with particle size, a increases with particle size. 4.3.4 Implication to fault development in the crust Because the earth's shallower crust is like a layer of moist granular material (see Chapter 3), the results of this study are applicable, to a first approximation, to crustal deformation. As discussed above, the upper crust is less cohesive, whereas the lower crust is more cohesive due to the confining pressure. This effect can be simulated by changing the particle size of the granular materials in the experiments. The results from the experiments and theoretical analysis both imply that in the uppermost layer of the crust, tensile structures should be common. With the increase of depth which accompanies the increase of confining pressure, shear structures will become dom inant. Large faults penetrate the whole crust of the earth. The whole crust is cohesive on average, which gives these faults predominantly shear mode. Under simple shear conditions, the shear generally takes the form of strike- slip faults; under compression, the shear may take the form of thrusts; and 92 under tension, the shear may take the form of normal faults. With such a point of view, shear becomes a general term describing parallel displacement along a fault plane in any direction without lateral opening displacement perpendicular to the fault walls. Thus the category of shear involves strike- slip as well as dip-slip faults. Typical tensile structure may only occur in rift zones which are characterized by high angle fault dip and apparent opening displacement between the fault walls. 4.7. Summary Experimental study of fault development in coarse fault gouge layer and theoretical analysis were performed to understand the effect of material cohesion on fault development and the relationship between cohesiveness and grain size of granular materials. The following preliminary results have been obtained from this study: a. Tensile fracturing takes place preferably over shearing in an incohesive granular material when it is subjected to simple shear. A through-going shear zone does not develop in such a granular material. b. Cohesion of a moist granular material is related to the particle size. Samller particles have larger adsorption force than larger particles because smaller particles have large specific surface area. Smaller particles partially saturated with water tend to make pores sealed in which low pore pressure can be developed upon dilation. All these factors make fine granular materials more cohesive than coarser ones. For a granular material consisting of equal size particles, the material becomes noncohesive if the particle size is larger than approximately 100 pm. 93 c For a sealed fracture developing in a granular material, internal friction angle of shear increases with grain size within the granular material. d. Because the earth’ s shallower brittle crust (the depth where groundwater can penetrate) is like a layer of incohesive granular material and the deep crust is like a layer of cohesive granular material, the experimental and theoretical studies can be used to explain fault development in the crust. Under simple shear condition, tensile faulting should occur in the shallower crust while shear structures should develop in the relatively deeper crust. For large faults penetrating both shallow and deep crust, shear dominates because the crust is cohesive as a whole. 94 Chapter 5 FRACTURE DISPLACEMENT MODE AND INTERNAL FRACTURE PRESSURE 5.1 Introduction Rock mechanics experiments indicate that under axial compression fracture growth from a preexisting fracture tip oriented in the optimum shear direction deviates from that direction, and soon becomes mode I growth parallel to the maximum compression (Nemat-Nasser and Horii, 1985; Ashby and Hallam, 1986; Sammis and Ashby, 1986). When a mode II fracture indeed develops at macroscopic scale under strictly controlled conditions, it is observed to consist of high density localized mode I microcracks at micro scale (Cox and Scholz, 1988; Petit and Barquins, 1988, Reches and Lockner 1994). It seems thus that shear fracture growth must involve tensile mode. However the experiments described in Chapters 3 revealed that under certain circumstances, mode II propagation may dominant: it has been observed that under simple shear plane stress, both naturally generated shear fractures and artificially inserted "preexisting" fractures can extend in plane. However it has been noticed that in-plane propagation is stable only when the material is relatively homogeneous and cohesive. Fracture development in a heterogeneous, incohesive material is still tensile (Chapter 4). In another experiment with fine gouge layer, it was observed that mode I propagation took place when excess water was available at a preexisting fracture tip (Fig. 5.1a) while mode II fracture develops when no excess water was available at 95 Fig. 5.1 A simple shear experiment with fine fault gouge layer (grain size less than 63 pm) showing that (a) tensile fractures extend from the lower ends of the "preexisting fractures" when extra water is available in, and (b) shear fractures start from "preexisting fracture" tips when no excess water is available. 96 the tip (Fig. 5.1b). The experimental technique was the same as that described in Chapter 2, except that in Fig. 5.1a every microslide was moistened before it was inserted in the layer while in Fig. 5.1b the microslides were inserted dry. Upon tilting the gouge layers, excess water within the "preexisting fractures" shown in Fig. 5.1a was observed flowing toward the lower tips of the "preexisting fractures" and accumulated there, but no excess water was observed at the fracture tips shown in Fig. 5.1b. It is geologically and mechanically important to explore this problem. While it seems that cohesion and excess water control the mode of fracture displacement, it really might be the pressure within a fracture which plays the role. Heterogeneous particle size can create large, percolated pore channels. As stated in Chapter 4, pore size is inversely related to capillary pressure as demonstrated. When pore size is large, capillary pressure is negligible so that fluid can flow freely within the channel. Mode I displacement involves expansion of the fracture opening. Because the expansion lowers pressure within the opening for an isolated sealed fracture, water preferably flows into the fracture (in case of heterogeneous granular materials). The water inflow can release internal pressure drop and thus mode I (tensile) fracturing can continue to grow under constant internal pressure. If, on the other hand, particles are fine and capillary pressure is high, water will be stagnated in the pores. This stagnated water blocks the connection between the fracture opening and its outside environment, therefore the fracture is isolated and the internal pressure drop retains. Tensile fracturing becomes unfavorable because extra energy must be spent to overcome the internal pressure drop. In this sense, water may only work as a medium either to transmit pressure to a fracture opening when the amount is sufficient (enough to form a 97 percolated system), or to seal the fracture from outside environment when the amount is insufficient. In order to establish a quantitative relationship between internal fracture pressure and fracture displacement mode, a theoretical analysis is carried out in this Chapter. By assuming that fracture mode is determined by the relative magnitudes of stress intensity factors for mode I and mode II, it can be demonstrated that internal pressure strongly affects the stress intensity factor for mode I (but not for mode II), and hence fracture mode. The result can be equally extended to faults. 5.2 Relationship between internal pressure and stress intensity factors The following analysis is for an isolated and sealed fracture. Specific consideration is given to pressure drop within a fracture caused by volume expansion of the fracture opening. 5.2.1 Derivation of the expressions for K\ Assuming plane stress, an isolated mode I crack of length 2 a subjecting to a confining pressure p, an internal fracture pressure pf, and tensile differential stress <r>0 (Fig. 5.2a). The stress intensity factor of the fracture is then: K, = o fn a Y (5.1) where o e is effective stress: < 7 , = 0 - P + Pr (5.2) Thus 98 t- k . k ( a ) •t cr (b) Fig. 5.2 (a) A tensile fracture of length 2a is subject to tensile stress a , confining pressure p and internal fracture pressure p/, (b) Flank displacement of a tensile fracture. 99 K, = (a ~ p + pf )(Ka)2. (5.3) Tensile fracture growth involves volume change. Adopting cylindrical coordinates and assuming the fracture is penny-shaped, then the volume of an idealized tensile fracture opening is (refer to Fig. 5.2b): V = HI xdxd(pdu (5.4) n where £ 2 refers to integration over a volume, 0 < x< a, 0 < (p < 2 n, and u is the relative displacement across a mode I fracture. According to Ewalds and Wanhill (1984), Supposing initial fracture length is a0 and its volume is V q , then after time t, the fracture length has increased to a and the corresponding volume to V. Then the volume ratio is: where a e 0 = a - p + pf0 is the initial effective stress (the pf0 is the initial internal pressure). For an isolated mode I fracture imbedded in an elastic media, the volume change causes pressure change inside the fracture. The volume and pressure are related through gas law: PfO^Q _ P f V where To and T are initial and current temperature within the fracture. Assuming isothermal condition (To=T) during fracturing, and using eqn. (5.7), we get The maximum u is Integrating (5.4) gives (5.5) y = 2 n a y _ 3 E (5.6) Yo=£ A V a ea3 ' (5.7) (5.9) 100 Using eqn. (5.2) to replace cr, in (5.9) and solving the equation for pp, we get Replacing pf in eqn. (5.3) with eqn. (5.10), we finally get the expression for Kp 5.2.2 Derivation of the expressions for Ku The K ji is derived by incorporating the term internal pressure p j in Melin's derivation (1986) of Ku. Plan stress is again assumed. An isolated fracture of length 2 a is assumed to subject to a confine pressure p , internal pressure pp and shear stress x (Fig. 5.3). Because of friction, mode II fracturing will not occur parallel to the applied shear stress r, but at an angle 6 to it. The basic expression for Ku is: In eqn. (5.13), T r is the resistant shear stress on the shear surface. Assuming Coulomb friction with friction coefficient p., the magnitude of xr is: (5.10) (5.11) K„ = xe(7ta)\ Where xe is effective stress parallel to the fracture plan: xe = t c o s 2 0 - xr. (5.13) (5.12) (5.14) where < 7„is the normal stress on the fracture surface: 0 if ( p - p f ) / r<sin20 p — P f -x s in 2 0 if ( p - p f ) / x > sin20. if {p-Pf)! x< sin20 (5.15) Bring (5.13), (5.14) and (5.15) into (5.12), the expression for Ku reads T(7ra)1/2c o s 2 0 i f ( p - p f ) / x < sin20 ^ U ’ i fn a ) [x(psin29 + cos2d)-pip-pf)} if (p-pf)! x > sin20, (5.16) 101 p 1 X X f. Fig. 5.3 Configuration of a mode II fracture of length 2a. 6 is the angle the fracture forms to the direction of the shear stress x, and p is the confining press. A mode II fracture will occur in the direction where Ku is maximum, which can be found in ^max “ ' if ( p- Pf) / T < 0 if 0 < (p-Pf) / X if (P-Pf)f x >p /(l +p z) with corresponding maximum stress intensity factor 0 -sin -1(p / x) if 0 < (p - pf) / X < p / ( 1 + j U 2) ^ . 2 J — \$ (r> — nI r n ! (\ 4- (5.17) 102 5.3 Discussions 5.3.1 Variations o f K\ and K u during fra ctu rin g Eqns. (5.11) and (5.18) give Kj and Ku as functions of fracture length, applied stress, confine pressure, and internal fracture pressure. Ku is sometimes a function of friction coefficient p.. In Fig. 5.4 the normalized Kf*=Ki a f n and normalized K u*=Kuafn (K*=K if ao=l unit) are plotted against normalized fracture length a/a0 . Parameters fi, p and pf are fixed as shown. The applied stress varies from 1 to 10 stress unit (specific unit is not given because of scale invariance). It can be seen in Fig. 5.4 that K[* decreases as an isolated fracture grows longer in mode I, while Ku* keeps increasing if the fracture is propagating in mode II. The difference between the K* and Ku* becomes larger as fractures grow longer. Actual fracture propagation occurs when Kj or Ku reaches a critical value called material toughness, Kjc or Kuc- At room temperature, Kic s for most rocks are between 0.01-4.0 MPa m1/2 (Atkinson and Meredith, 1987). Limited data indicate that for some rocks the Kic increases with confining pressure but decreases with temperature (Barton, 1982). The study on Kuc is at infancy stage at this time. Available data seems to indicate Kuc on the order of 1 MPa m1/2 (Li, 1987), which gives ratio Kuc/Kic between 0.25-100. mu=0.65, p=10 pf=10 70 60 c4 0 30 20 a/ao Fig. 5.4 Plot of K * and Ku* as functions of normalized fracture length a/aO by assuming n=0.65, p=10 unit and pp= 10 unit. The dashed lines represent Ku* and the solid lines represent K*. The applied stresses vary from 10 to 100 unit. Note that Ku* is much higher that Ki* under the assumed conditions. The Kuc/Kic can be further constrained by converting critical energy release rate Gc to Kc. Laboratory derived Gc's for rocks fracturing in mode II are generally of the order 104 J n r2 (Li, 1987), while Gc's for mode I fracturing are mostly between 101 to 102 J n r2 ( Atkinson and Meredith, 1987). For ju =15-30 GPa, v=0.25 as typical shear modules and poisson's ratio for most surface rocks, and using equation G =(l-v)K2/2|i (Li, 1987), G//c=104 J m2 translates to Kuc =20 MPa m1/2, whereas G/c=10-102 J nv2 translate to K\c = 0.6-2 MPa m1/2. 104 Therefore Kuc/Ku ratio ranges from 10-30. In Fig. 5.4, all Ki*'s decrease as fracture length increasing, with those at lower stress levels approach zero fast as expected, and Ku*'s keep increasing with fracture length. Kuc/K[c reaches 10 for most stress levels when fracture length is quadrupled (marked by a vertical line in Fig. 5.4) and the ratio keeps increasing thereafter. Mode II fracturing thus surpasses mode I and becomes a dominant mode. It should be pointed out that without pressure effect both K[* and Ku* would increase with fracture length a . The decrease of K[* with a is solely p=pf vary, mu=0.65, tau=sigma=10 80 70 K I* K ll* all p 60 c 40 30 20 20 a/a0 Fig. 5.5 By fixing the applied stress a and x to 10 unit, and letting p=p/vary, it shows that the Ku* becomes the same for all p=p/ while the Kj* decreases with the increase of confining pressure p=p/ (internal pressure). Friction coefficient p. is still 0.65. 105 due to internal pressure, since it is negatively proportional to a in (5.10). Fig. 5.4 indicates that the effect of the pressure drop in a mode I fracture is so strong that it overrides the effect of length increase to make K * decrease. K f drops very fast at the beginning, suggesting that for imbedded, and isolated cracks, its growth under mode I may be a short-life event. In Fig. 5.5, we keep a = r=10 unit, but p = pf vary from 0 to 20 unit. In this figure, Ku*'s are represented by a single line which matches the line indicating Kj* at p=0. Note that confining pressure p has no effect on Ku* because p =pf. Kj*, on the other hand, keeps dropping as p increase. Note that when p is greater than 12 unit, all K*'s approaches zero, meaning mode I fracturing is impossible. The implication of this finding is that what we frequently observed tensile fractures on the earth's surface may not be so common in the depth, except when fractures are open systems in which case fractures can somehow regain their internal pressure (e.g., air inflow, fluid percolation and magma intrusion, etc.). Otherwise, shear fracturing should dominant in the depth because by assuming p-pj constant (in Fig. 5.5, p-pf= 0), Ku* is independent of p and keeps increasing with a . Melin arrived the same conclusion in 1986. Above plots assumed initial p=pf. In case p^pp two typical examples are given in Fig. 5.6 and Fig. 5.7. In Fig. 5.6, we assumed p«pf. This results in K[*>Ku* at first and mode I fracturing takes place. However, the situation changes very soon. Kj* drops sharply with fracture length increasing. Around a~3ao, Ki* becomes equal to Ku*- After this turning point, Ki* becomes less than Ku*. It is noticeable that some of the fC /*’ s with higher stress levels drop very quickly at first with increasing a/ao, then slow down and turn back, leaving a minimum value at about a=3ao. Also note that even Ki* increases 106 mu=0.65, p=10 pf=20 80 70 60 50 c 40 30 20 a/ao Fig. 5.6 At high internal pressure (p/=20 unit) arid low confining pressure (p=10 unit), tensile fracturing is preferred at beginning due to high Kj*, but becomes unfavorable as fracture length increases. with a/ao after the turning point, it is still less than Ku* by as much as nearly 10 unit in magnitude. Fig. 5.7 shows a situation where p is much higher than pf. Most K *'s are close to zero and eventually becomes negative as the length increases. The Ku*'s increase with length as usual. The difference between K[* and Ku* is large for the two types of fractures. For example, when a= 2 0 a o and stress level at 10 unit, Ku* equals to ~50K/ * . Therefore mode II fracturing is preferable. 107 Fig. 5.7 At high confining pressure (p=10 unit) and low internal fracture pressure (P/= 1 unit), Ku* is much higher than Ki*. Since most Kj*'s are zero or negative, only mode II fracturing is possible. Finally, the effect of fi on Ku can be seen by comparing Fig. 5.7 and Fig. 5.8. The reduction of p. from 0.65 in Fig. 5.7. to 0.1 in Fig. 5.8 almost doubled the magnitude of Ku*. However, the effect is important only when confining pressure p is higher than internal fracture pressure p/. In case p<p/( the p has essentially no effect on Ku*. Fig. 5.8 Friction coefficient n has some effect on K jj* as can be seen by comparing this figure with Fig. 5.7. All the other parameters are the same except /i for both figures. 5.3.2 Implications The implication of the study is that tensile fractures and faults should be more popular near the earth's surface than in the depth. One reason is that a fracture (or fault) system near the earth's surface is more likely to become open system (or percolated) because it can cut through the earth's surface, or can be penetrated by underground water circulation. This helps maintain a constant internal pressure within the fracture during its growth, and therefore maintains mode I propagation. At lower depth, each fracture or fault tend to be isolated, and water circulation stops. Mode I displacement therefore becomes less possible as the internal pressure drops. Another reason is that confining pressure increases with depth. As can be seen in Fig. 5.5, under lower confining pressure, the magnitudes of Ki and Ku are close, hence mode I displacement is preferred. Under high confining pressure, K\ is much lower than Kjj, then mode II displacement prevails. There are some processes that can make mode I displacement a favorable mode in the deep depth, though. For example magma intrusion can generate high internal pressure as well as a large amount of fluids. These fluids and magma itself work their way into fractures and faults, make pf high than p, and drive the fractures and faults to grow in mode I. In The Geysers geothermal field in northern California, the most productive fractures are tensile fractures which are in horizontal (Beall and Box, 1989; Thompson and Gunderson, 1989). High P-T vapor comes from circulating underground water which is heated by an intrusion body underneath the region. The tensile fracturing makes the whole region expands, but confining pressure also becomes higher which prevents further expansion. Then the only dimension it can expand further is toward the free boundary— the earth's surface. Therefore horizontal fractures are extensively developed and become the most productive structure. 110 5.4 A model for in-plane propagation of shear fractures in fine granular materials Based on this study, a model for in-plane shear propagation of shear fractures is proposed. First let us consider the situation in the clay and fine gouge used for the experiments. The clay has the smallest particle size (<2 p) and fine gouge has the moderately small particle size (<125 p). When these materials are partially saturated with water, numerous pores are formed. Based on the result from Chapter 4, these pores are small and had a high capillary pressure. The high capillary pressure has two effects: one is to pull particles together to make the granular material more cohesive, and the other is to stagnate water drops in the narrowest throats between pores to seal those pores (Fig. 5.9a). Most, if not all, pores in the moist clay and gouge are therefore sealed. When such a material is subject to a simple shear stress, the sealed pores inside first dilate to the limit of Pp+ gt =P (19) where Pp is pore pressure, crr the dilation stress and P the confining pressure. In the above equation a T and P are constants but Pp is a variable which decreases in value with the pore dilation. After the balance in eqn. (19) has been established, further dilation of pores and tensile fracturing is not possible. This is the initial dilation hardening. While dilation and tensile fracturing are stopped, shear stress is not affected much by the dilation. Moreover, dilation of pores increased pore volume and decreased contact area between grains, which in turn decreased 111 (C ) ( d ) Fig. 5.9 A suggested mechanism for in-plane propagation of shear fractures in granular materials, a) illustrates pores are sealed by stagnated water, b) shows a shear (S) nucleated by grain boundary sliding and coalescence of sealed pores after dilation hardening, c) shows after nucleation, shear propagation continues in front of a shear (S) tip through grain boundary sliding over the sealed pores, and d) demonstrates that excess water forms a percolate system to make pores no longer sealed and therefore tensile fracture (T) develops. 112 friction between grains. Grain boundary sliding takes place along the maximum effective conjugate shear directions. Many sealed pores are coalesced during the grain boundary sliding, forming embryonic shears (Fig. 5.9b). Numerous dilated pores created numerous embryonic shears that are densely distributed all over the sample. Because shear displacements along such shear structures are very limited, they appear only as protofractures on a sample surface. The subsequent evolution of the protofractures into strike-slip fractures is a competitive process. Since many embryonic shears have been developed simultaneously, some of the shear strain has been released. The remain shear strain is accommodated by a few larger shears emerged from competitive growth. The competition ability of each embryonic shear depends on its scale (larger shears are more competitive), mechanical property of the material and layer thickness (more elastic and thicker material would retain fewer fractures), a similar mechanism used to explain tensile fracture spacing by Price (1966), Hobbs (1967), Sowers (1973), and Banerdet and Sammis (1992). The final surviving shear structures are those whose scale and spacing from each other are appropriate to accommodate the shear strain. The remain embryonic shears between them are disabled. Now the propagation from their tips of the survived fractures follows a similar process as during shear nucleation, i.e., grain boundary sliding and coalescence of the sealed and dilated pores (Fig. 5.9c). Under a relatively dry condition (no excess water), the shear propagation will continue even when the body of a shear structure is no longer sealed. However under the condition that excess water can get into a fracture opening, the propagation will change to mode I because 113 excess water can form a percolated system in front of a shear fracture tip so that pressure drop within the pores is released (Fig. 5.9d). The most important condition for these to happen is that pores must be sealed, meaning there is no material exchange between a pore and its external environment. Otherwise the balance showing in eqn. (19) would not happen, and tensile propagation of fault would happen. A similar process might take place in the crust. Except the very shallow crust (within the depth groundwater can penetrate) where faults are not sealed and therefore might propagate in mode I, the relatively lower crust has two situations. The upper brittle layer is granular which contains ground water, a situation similar to the moist coarse granular materials used in this study. The lower layer is ductile and cohesive due to temperature increase. Confining pressure increases downward. All these situations favor the development of a closed system for a fault. Because opening displacement tends to be suppressed by confining pressure as the internal pressure within a fault drops, the fault should propagate as shear. 5.5 Summary To summarize, it has been recognized that internal pressure plays an important role in controlling fracture displacement mode. During a fracturing event, internal pressure in a tensile fracture (or a mode I fracture) tends to decrease due to volume expansion. If the fracture forms a closed system itself (i.e., isolated and sealed from outside environment), the pressure drop can not get balanced with the confining pressure and it will eventually become large enough to stop further fracture propagation in mode 114 I. However, if a fracture forms an open system (i.e., it permits air and fluid penetrating), the pressure drop can be released and mode I displacement can sustain. Sufficient water and heterogeneity enhance mode I displacement. On the other hand, typical mode II displacement (shearing) doesn't involve as much volume change as mode I displacement does, thus pressure within a shear fracture is about constant and K jj always increases with fracture length under constant stress, no matter the fracture is isolated or not. If mode I displacement becomes unfavorable because of internal pressure drop, mode II displacement will take place under appropriate shear condition and become a dominant mode. High confining pressure promotes mode II displacement. Friction has little effect on the change of fracture displacement mode. A model for in-plane propagation of shear fractures is proposed based on this theoretical study. 115 Chapter 6 ORIGIN OF EN ECHELON STRIKE-SLIP FAULTS 6.1 Introduction Field observations (Crowell, 1974; Segall and Pollard, 1980; Deng et al. 1986; Sibson, 1986; Wesnousky, 1989) and laboratory experiments (Brace and Bombalakis, 1963; Peng and Johnson, 1972; also Chapter 3 of this study) indicate that a strike-slip fault consists of many discontinuous segments in en echelon. The echelon arrangement is simple at fracture scale: micro fractures within the trace of a shear fracture follows exclusively the rule that under left-lateral shear stress configuration the micro fractures arrange into right- stepping arrays while under right-lateral shear condition, the micro fractures arrange into left-stepping array. The arrangement is more complicated at fault scale: fractures and fault segments within a strike-slip fault step both left- and right-laterally within a single fault. The unique arrangement of micro fractures within a shear fracture leads to the proposition that the micro fractures may be tensile fractures because their orientations are close to the tensile plane (Brace and Bombalakis 1963, Peng and Johnson 1972, Knipe and White 1979, Petit and Barquins 1988, Cox and Scholz 1988, Reches and Lockner 1994). This idea seems to be justified by the consideration of stress intensity factor: the critical stress intensity factor for shear propagation, Knc, seems one to two orders of higher than that for tensile propagation, Kic (Wang 1982, Li 1987). Therefore it follows that a shear fracture should start with a series of tensile micro 116 fractures by spending less energy, and then the tensile micro fractures link up forming a macroscopic shear fracture (Peng and Johnson 1972, Petit and Barquins 1988, Roches and Lockner 1994). If this is the case, then all the micro fractures should arrange into en echelon and follow the unique stepping rule as described above. However, the tensile fracture mechanism can not explain the coexistence of both right- and left-stepping of shear segments in a strike- slip fault. Furthermore, experiments described in Chapter 3 indicate that the micro fractures developed in clay and gouge layers are actually shear fractures instead of tensile ones. The orientations of the micro shears are different in different shear sets but they always tend to be approximately parallel to the orientation of the macroscopic shears. Therefore some different mechanisms might control the process. In this chapter, the origin of en echelon shear structures is studied experimentally (including using some of the experimental results from previous chapters) as well as numerically with finite element analysis. The study revealed that there are at least three other mechanisms for the development of echelon shear arrays: 1) mutual shearing offseting of different shear sets, 2) coalescence of shear structures, 3) mutual avoidance of collinear shears. 6.2 Experiments Although the basic experimental procedure was the same as that described in Chapter 2, some specific changes were made to fit the specific purpose of this chapter. Three slightly different simple shear experiments were performed in this study. In experiment one all structures were 117 generated naturally in the same way as those described in Chapter 3. The difference from those in Chapter 3 was that in this experiment a higher water content (43wt.% compared to 39wt.% for most samples used in the other experiments) was kept in a fine fault gouge layer (grain size smaller than 63 g) and the sample layer was made thinner (2 cm). This higher water content was found suitable for producing well-developed patterns of echelon shear arrays. The experiment was designed to monitor the dynamic evolution of echelon shear arrays. Experiment two had preexisting fractures prepared collinear along the two conjugate shear directions (Si and Si', refer to Fig. 6.1a) in a fine gouge layer (grain size was smaller than 62.5 gm and water content was about 39wt.%). The preexisting fractures were prepared by inserting microslides, 6 cm long and 4 cm wide, into the gouge layer. Because a sample rotated during experiment and so do the shear structures, the preexisting fractures were not inserted until the Si and Si' protofractures (which represented the embryonic shear structures) emerged. Within a collinear preexisting shear array, the distance between two adjacent shears was about 3-5 cm. The purpose of the experiments was to examine whether collinear shear fractures link up directly or they avoid each other. Experiment three used the same material and prepared preexisting fractures in the same way as in experiment two, but this time some linear markers were inscribed on the sample surfaces before experiment to represent another shear set (Fig. 6.2). The purpose of this experiment was to examine if the shear activity on one shear set could make another set (represented by linear markers) echelon. 118 6.3 Experimental observations In the experiment one which did not involve preexisting fracture in a layer having higher water content (43wt.%)/ shears nucleated and propagated in the same way as those described in Chapter 3. Long and dense protofractures appeared in conjugate first when strain reached ~0.35 (Fig. 6.1a). As strain increased further, some protofractures acquired appreciable offsets and evolved into macroscopic shears, while other protofractures were disabled as a result of stress relaxation (caused by shear stress release along the active shear fractures). At these early stages, however, few shear structures showed en echelon segmentation (Fig. 6.1a). As strain increased further, each shear fracture rotated and the aperture was widened. Within each shear fracture, micro fractures gradually showed up which were arranged in echelon, making the macroscopic shear structure feather-like (Fig. 6.1b). Under this right-lateral shear condition, all the micro fractures in the Si shear set made left-stepping to form en echelon arrays, while all micro fractures in the Si' shear set made right-stepping to form echelon arrays. The echelon micro fractures were identified as shear fractures for reasons that they were almost parallel to the strike of the macro shear structures, and were different in different shear sets. At the same time, it was found that between any two en echelon micro fractures in the bridge area there was usually a small fracture that cut an original shear fracture almost perpendicularly. The types of the fractures were identified to be the conjugate shears Si (which cut Si' set) and Si' (which cut Si set) based on their orientation and shear displacement. The echelon micro fractures became more visible as strain increased further. 119 Fig. 6.1 Echelon shear arrays developed in a gouge layer with high water content (42wt.%). Applied simple shear is shown by the arrows, (a) shows the initial stage at which straight protofracture (embryonic fractures) cross each other. No en echelon array is observable; (b) shows that as strain increases, the conjugate shears cut and offset each other, creating echelon shear arrays. M O In the experiment two where preexisting fractures were made collinear with overall orientations parallel to the maximum conjugate shear directions, fracture propagation path was observed to be related to the collinear separation between two neighbor fractures. If two collinear fractures were separated less than 3 cm (experiments show that this separation distance varied with layer thickness and mechanical property), the propagation of the fractures was parallel to the preexisting shear plane and thus the two fractures linked up directly end by end (the near horizontal array in Figs. 6.2a and b). However, if two fractures were separated more than 4 cm they propagated several degrees apart from the preexisting shear plane (the near vertical array in Figs. 6.2a and b) forming echelon arrays. The stepping of the fracture segments within a shear array (a fault) was uniquely related to shear sense: if the shear sense was right-lateral, left-stepping happened, and if the shear sense was left-lateral, right-stepping happened. It was also noticed that there was a transitional period at the beginning of a fracture propagation during which the new fracture growth was tensile, but it soon turned back to shear propagation. As the strain continued increasing, a second set of shear Fig. 6.2 (next page) A gouge layer experiment with microslides inserted along the two optimum shear directions to represent preexisting shear fractures. Applied simple shear force is shown by the arrows. In (a) preexisting fractures start to grow from the preexisting fracture tips. Note that when the initial fractures are separated far apart (see the near vertical array), fractures don't grow toward each other. However, if the fractures are close (the near horizontal array), they link-up end by end. In (b) the shear fractures in the near vertical array avoid each other initially forming echelon array, and then link up by through newly developed fractures in the bridge area. 121 122 structures developed rapidly at the original preexisting fractures tips (arrows in Fig. 6.2b) and linked up the preexisting fractures directly. In the experiment three in which linear markers were inscribed (Figs. 6.3a and b), the linear markers were dislocated by the shear displacement on the preexisting fracture arrays. The original continuous linear markers then became echelon. The stepping was also unique and dependent on the shear sense as described in experiment one. The linear markers were used to represent a conjugate shear set. Note that both primary synthetic shear Si (Fig. 6.3a) and its conjugate Si' (Fig. 6.3b) could tailor the linear markers into echelon geometry. The link-up of the individual shears was the same as those shown in Fig. 6.2. 6.4 Finite element analysis A 2-D linear elastic finite element analysis was performed to examine stress state in the vicinity of several interacting shear fractures and the effect of shear displacement on fracture geometry. The analysis was performed by using a software COSMOS developed by Structural Research and Analysis Corporation. The material where fractures were hosted was assumed to be granite with Yung's modules equal to 4xl01 0 Pa and Poission's ratio of 0.22. Shear forces were applied on the boundaries as shown in Fig. 6.4a. The lower boundary was fixed to generate simple shear. In the first analysis, two collinear mode II fractures were introduced which were initially separated horizontally by a distance L. In the second analysis, a conjugate shear fracture was introduced between the two collinear fractures in the first analysis. The 123 Fig. 6.3 Linear markers inscribed on a sample surface are offset by primary shear Si in (a) and conjugate primary shear Si' in (b). Simple shear direction is shown by the arrows. N > (a) Yung’ s modules: 4x1010 Pa; Poission's ratio: A... Fig. 6.4 Finite element analysis of the interaction of shear fractures. The loading and boundary condition is shown in (a) and shear stress is shown in deflected manner in (b) and (c). In (b) only two co-parallel fractures are involved but in (c) a conjugate shear is also introduced. Note the anti-clockwise rotation of shear fractures in (b) and lateral offset of the collinear fractures by the conjugate fracture in (c). Fig. 6.4 (continued) ■ M B B B H T a u XV 5 . 6OE+03 4 . 8 7E -*-03 3 .40Ei-03 4" -2.G 7E+03 l .9<JE+03 I .21 Ei-03 478 . 8 0 0 0 - 2 5 3 . 0 0 0 Tau_ XV r6 .45E*03 |~5 .6QE+0 3 ~4 . 7 6 E * 0 3 h 3 .92E1-03 £ ^ ~ 3 .8 7E*03 $/ / * * v B S k -2 . 2 3 E * 0 3 I . 38E -*-03 539.8000 - 3 8 5 . 0 0 0 second analysis was considered to be more realistic because shear usually occurs in conjugate. Figs. 6.4b and c show the results. Fig. 6.4b shows that shear stress between the two interacting collinear fractures in an elastic material is the maximum along the direction defined by the two fracture tips. Therefore if fracture propagation ever happens, the two fractures may eventually link-up end by end. It is also noticed that the two fractures rotate anticlockwise during the deformation about their own axes going through the centers of each fracture. If there is any possibility that echelon arrangement occurs, the stepping would be right-lateral in this right- lateral shear condition, which is in conflict with laboratory observation shown from Figs. 1 to 3. Fig. 6.4c shows that after a conjugate shear fracture has been introduced between the two collinear shear fractures, the shear stress between the two fractures becomes discontinuous, and the two fractures are shifted out of the same shear plane by the lateral displacement along this newly introduced conjugate shear fracture. The two originally collinear shear fractures are now in echelon. Under this right-lateral shear condition, one fracture stepped left- laterally with respect to the other. This stepping is consistent with the experiments shown in Figs. 1 through 3. Also note that because of this newly introduced conjugate shear fracture the collinear fractures no longer rotate. In brief, the analysis indicates that two collinear fractures would link up end by end if there is no conjugate shear fracture between them, and they become echelon if there is such conjugate fractures. This analysis agrees with the laboratory experiments described above. 128 6.5 Models Three models are proposed for the development of echelon shear arrays based on the experimental observations and finite element analysis. The first one is a comprehensive model which includes mutual shearing offsetting of two intersecting shear sets and subsequent growth of the shear segments; the second one is the coalescence of different shears during their interaction, and the third one is the mutual avoiding of collinear shears. Model 1: M utual shearing offsetting The model includes nucleation of shear structures (initially as protofractures), mutual shear offsetting of the shear sets, continuous growing of the segregated shear segments in plane and final reunion. These processes are explained step by step in the following. 1. Nucleation of shears: Long and straight protofractures develop from coalescence of defects (mainly pores). Two (or more) sets of the conjugate protofractures form a shear grid. The protofractures do not contain echelon segments at this time (Fig. 6.5a). 2. Mutual shearing: After a period of competitive growth, some of the protofractures grow into shear fractures. Once appreciable shear displacements have occurred on the active conjugate shear sets, the shears in one set cut the shears in the other set into small segments (Fig. 6.5b). Due to the shear offset along each of the shear sets, an originally continuous and straight shear trace now becomes an array of echelon shear segments. For the right-lateral shearing, the shear segments step left in synthetic shear set and step right in the antithetic sets. If the shear sense is left-lateral, the step directions are reversed. The mutual shearing also causes rotation of each 129 t I t (a) l (b) f l f I (c) (d) Fig. 6.5 Idealized process leading to the development of en echelon shear arrays by mutual cutting of shear sets and simultaneous growth, (a) shows the coalescence of the microshears forming embryonic shears (protofractures), (b) illustrate segmentation of shear structures by conjugate shears and (c) shows the new growth of the offset shear segments. Echelon arrays are fully developed at this stage, (d) shows another episode of coalescence. 130 shear structure as a whole: the synthetic shears rotated several degrees anticlockwise, and the antithetic shears rotated several degrees clockwise (Fig. 6.5b). 3. Growth: Each shear segment in a shear fracture, now cutoff at both ends by two conjugate shears, must break through the bends formed by the conjugate shears in order to grow. As they grow in plane, overlap occurs between two shear segments, and interaction between the two shear segments becomes stronger (Fig. 6.5c). 4. Reunion: The interaction between the shear segments as they become close and develop overlap causes segment reunion. Different segments which originally belonged to one macro shear and late were segmented are reunited, forming a through-going shear (Fig. 6.5d). The coalescence is essential for developing a macro shear to accommodate large shear strain. This reunion smoothed out bends formed by mutual shearing and a fracture trace again becomes simple. A new cycle of shear segmentation and reunion will begin. As fractures coalesce forming faults, the process moves on to fault scale. Each time it causes the rotation of the overall shear trace and creates more gouge materials by continuous wearing (each reunion cause breakdown of the bridge areas between shear segments). An example of the mechanism may be found in Southern California between the San Andreas and San Gabriel fault system (Fig. 3.12). The San Gabriel fault was once part of the San Andreas fault system and was abandoned since Pleistocene (Crowell, 1974). The new route of the San Andreas fault and the abandoned San Gabriel fault form a fault loop. Using the model proposed here, the evolution history of the structure can be explained as following (Fig. 6.6): The San Andreas fault (represented by fault 131 i (a) (b) (c) Fig. 6.6 A possible mechanism for the development of San Andreas-San Gabriel fault loop, a) shows strike-slip fault A is offset by a conjugate fault B; b) shows the two cutoff ends of fault A break through B to grow; c) illustrates that one branch of A reunions with the other branch and a fault loop is formed. A in Fig. 6.6) was once cut across by an east-west conjugate strike-slip fault (represented by fault B in Fig. 6.6) into two segments. The left-lateral displacement on the fault offset the northern segment toward west and the southern segment toward east (Fig. 6.6a). The two cutoff ends of the San Andreas fault thus became "dead" ends and worked as new fault tips. Stress concentration in the "dead" ends eventually broke through the conjugate fault (Fig. 6.6b). The northern end of the southern segment propagated toward north forming the new extension of the San Andreas fault which eventually reunited with the northern segment of the San Andreas fault, while the southern end of the northern segment propagated toward south and eventually joined the southern segment of the San Andreas (Fig. 6.6c). The San Gabriel fault was disabled after this reunion. The same process is 132 now undergoing in Big Bend area where the Garlock fault intersects the San Andreas fault. However, large scale shear structures can not be fully explained by this model. For example both left- and right-stepping of the shear segments have been observed in natural strike-slip faults but above model only predict one stepping style in one shear fracture or fault. Therefore another model is need which is suggested to be fracture coalescence, as to be discussed below. Model 2: Coalescence This mode is based on experimental observations described in Chapter 3. One of the typical fault patterns is shown here in Fig. 6.7 for easy reference. In this pattern both left- and right-stepping of fracture segments were observed due to fracture interaction and coalescence. Fractures involved in the coalescences include independent shear fractures as well as shear segments which originally belonged to one shear fracture but were segmented late by other shear set (arrow in Fig. 6.7). The model suggests that if a population of discrete fractures develop in a region (Fig. 6.8a), mechanical interaction happens when fracture density becomes high. The result of the interaction is fracture coalescence--link-up of several smaller fractures into one larger compound fracture— strike-slip fault. It has been reported in Chapter 3 that there are several styles of fracture coalesce (Fig. 3.5). If several Si fractures are linked up through Si' shears or T structures, a right-stepping echelon shear array forms; if they are linked up through S2 shears, a left-stepping shear array forms; if they are linked up by utilizing both Si' (or T) and S2 fractures, both right- and left-stepping of shear segments occur in the same shear zone (Fig. 6.8b). The coalescence of two 133 Fig. 6.7 Echelon shear arrays developed mostly by coalescence in a gouge layer with less water content (39wt.%). Also note the mutual offset of conjugate shears (arrows) and the development of pull-apart basins (holes). Both right- and left-stepping lead to the development of pull- apart basins. Simple shear is shown by the arrows. 134 : ! ■ M (a) (b) Fig. 6.8 Sketch showing the development of en echelon shears by coalescence, (a) shows discrete faults before coalescence, and (b) shows an en echelon shear array (both left- and right-stepping) developed after coalescence. shear fractures is not simply end by end, but usually the two fractures overlap first and then grow together (Fig. 6.7, also Du and Aydin, 1993). Generally only one end of a fracture join with another fracture, the other is left dangling because the stress in the interaction area has been released upon the joining of the other fracture tip. This gives the whole faults an echelon appearance (Figs. 3.5 and 6.7). Model 3 Mutual avoiding of collinear shears The model shares many similarities with model 1. But it deals with the interaction of collinear shear structures (fractures or faults). The model propose that an array of collinear shears avoid each other during propagation and become echelon because of: 135 ti w (a) (b) (c) Fig. 6.9 Development of echelon shear arrays by mutual avoiding of collinear shears. The avoiding can be caused by conjugate shearing (a), plastic deformation (b) and tensile fracturing (c). 1. Conjugate shearing: conjugate shear fractures usually exist between two collinear shears. The shear displacement on a conjugate shear fracture offsets laterally the two collinear shears out of the same shear plane and thus they become non-collinear (Fig. 6.9a). The far the separation between the two collinear shears, the more conjugate shears are likely to be involved, and the more shear offset is expected. Two very closely-spaced collinear fractures link up directly without forming echelon geometry due to strong interaction and less chance to have conjugate shears in between. 2 Plastic deformation: for some more ductile materials, shear fractures may not develop between the two collinear shears but plastic shear deformation can occur (Fig. 6.9b). The effect is similar to the brittle conjugate shearing. 3 Tensile propagation: if tensile propagation is involved in front of a preexisting shear fracture tip as observed in Fig. 6.2, then a collinear array of 136 shears will grow into echelon, even though the tensile propagation is transient and shear propagation can resume later (Fig. 6.9c). The propagation mode is mostly controlled by internal fracture pressure which has been discussed in Chapter 5. All those three factors make collinear shears avoid each other and develop into echelon geometry. 6.6 Summary Experimental observations and computer modeling lead to the proposition of three models for the origin of en echelon shear arrays. The first model suggests that an en echelon shear array stems from a comprehensive process which involves initial development of continuous shear sets, later segmentation and offsetting of the shear sets each other, and finally reunion of the shear segments. The process is cyclical and happens in all scales. The second model suggests that coalescence of discrete shears during their mechanical interaction leads to the development of echelon geometry. And the third model suggests that an array of collinear shears would avoid each other during fracture growth due to conjugate shearing, plastic strain and possibly tensile displacement, developing into echelon geometry. The second model is most applicable to development of large scale shear structures. 137 Chapter 7 A CELLULAR AUTOMATON FOR THE DEVELOPMENT OF CRUSTAL SHEAR ZONES 7.1 Introduction The former chapters described experimental study of fault development under simple shear. Important observations have been made about fault nucleation, propagation and interaction. With the faulting parameters constrained from the experiments, it is possible to build a computer model to simulate a fault pattern evolution. The model can serve at least three purposes: it can confirm the parameters which control the evolution of a shear zone, it can be used to constrain some other faulting parameters which can not be well constrained by the experiments, and it can show step by step the development of a fault pattern. In this chapter, a computer model is formulated for these purposes. Development of the computer model is guided by experiments described in the former chapters. The model parameters are constrained by direct comparison with natural fault patterns. Computer simulations of distributed faults generally follow one of two approaches: mechanical models and cellular automatons. Mechanical models involve the progressive deformation of a network of elastic springs or beams (Takayasu 1985, Herrmann et al. 1989 Meakin 1989) in which the spatial distribution of elasticity and/or strength are preassigned. Each time an element is broken, the stress distribution throughout the network is 138 recalculated and the next element to be broken is determined. Although such models simulate the elasticity of the crust, they are computationally intensive, and thus are limited to arrays which are too small to model the evolution of large scale crustal shear zones. The second type of computer simulation is the cellular automaton in which a few simple rules guide the evolution of the system (Toffoli and Margolus 1987, Bak and Tang 1989). The failure of a cell and redistribution of stress is determined by rules which are based on the results of experiments and theory in elastic fracture mechanics. Because the method is simple and fast, the modeling array can be very large. In the next section, we define our automaton for the growth of a network of strike-slip faults and establish the rules which guide its evolution. 7.2 A cellular automaton for the nucleation, growth and interaction of strike- slip faults The cellular automaton consists of a 2-D array of equal sized square cells. A cell can be either intact or broken. An intact cell is assigned a value of "0" while a broken cell is assigned an integer which equals the fault length (i.e., the number of contiguous broken cells). In order to minimize boundary effects, the automaton has periodic boundary condition in both x and y. The displacement and the subsequent modification of a fault trace by wear were not taken into account in this model. The evolution of a fault pattern on this array is controlled by three sets of rules which govern (1) the nucleation of new faults on the array, (2) the growth of established faults, and (3) the interaction between faults leading to their coalescence. It has been assumed 139 that the array is being deformed in right-lateral simple shear which is incremented into equal time steps. In the following, the three sets of rules that guide the evolution of the automaton are described. 7.2.2 Nucleation of faults on the array There are two decisions which must be made in specifying the nucleation of new faults on the array. First we must determine how many new faults are to be nucleated in each time step and then we must determine where each nucleation is to occur. The number which are nucleated during each time step is determined through the artifice of following the growth of a population of "starter fault seeds" (e.g., fractures, grain boundaries, and other types of defects), all of which are initially smaller than the cell size of the automaton. Each starter fault seed increases in length during each time step. The number which reach the size of an automaton cell in each time step determines how many new faults are nucleated during that interval. It is important to note that no position is assigned to a starter seed, only a length. The position of the nucleated faults on the array are chosen at random, and therefore, no spatial fractal structure is preassigned. The artifice of using such an array of starter fault seeds allows the author to follow a wide spectrum of fault seed sizes which control nucleation without requiring an intractable number of automaton cells. The distribution of starter seeds is assumed to follow a power-law distribution since the statistical analyses of fracture and fault length have 140 been observed to be power law (Nur 1982, Segall and Pollard 1983a, Sammis et al. 1992). A distribution of the following form is assumed N(L)=CL'm (7.1) where N is the number of faults of size L , and C and m are constants. Starter fault seeds are assumed to grow at a rate which is proportional to their length. This is the same rule used to control the growth of faults on the automaton array. It can be seen below that it is justified by the observation that it yields the most realistic fault patterns. Interaction between the growing starter seeds is ignored. For convenience, a discrete size distribution of starter seeds is used in which the size changes by a factor of 2. Eqn. (7.1) can then be rewritten as: N(L)=CL-log 2B (7.2) where B is related to m by B=2m (7.3) The parameter C is chosen so that only a few faults are nucleated at the initiation of a pattern and fault growth is not swamped by nucleation. 7.2.2 Growth of faults Once nucleated, faults on the array grow by two mechanisms: in-plane extension and coalescence with other faults. The rules for the in-plane extension including propagation direction and rate are discussed here; those which control coalescence are discussed in the next section. 141 There is some controversy as to whether shear fractures can grow in their own plane. Laboratory experiments on cracks in elastic media (Brace and Bombolakis 1963, Lajtai 1971, Ingraffea 1981, Horii and Nemat-Nasser 1985, Ashby and Hallam 1986, Sammis and Ashby 1986) observe that loading of preexisting cracks results in the nucleation and growth of out-of-plane tensile fractures at the crack tip. However shear cracks in granular media are observed to extend in plane (Reches 1988, also Chapter 3), as are shear fractures in triaxial experiments at elevated confining pressure (Reches and Lockner 1994). The in-plane propagation appears to be the result of a suppression of the out-of-plane tensile fracturing by pressure drop within a fault or confining stress (Chapter 6). The observation of long shear faults in the crust argues that, at the large scale, in-plane extension is also favored under crustal conditions of fluid pressure and confining stress. A second outcome of the experiments described in the former chapters is an automaton rule for the orientation of the faults. It is commonly believed that a strike-slip in a simple shear field should be parallel to the simple shear direction. However the experiments do not support this a priori speculation. First it has been observed that there are several shear sets developed in the emerging fault pattern, only one of which (S2) is parallel to the simple shear (Figs. 3.1 to 3.3, and Fig. 7.1a). The author is left with the choice of which set to simulate. Since the primary shear set Si was observed to lead to the formation of a through-going shear zone (correspond a large scale strike-slip fault), the set is chosen. The other sets were omitted in the automaton for simplicity but their importance in fault coalescence will be emphasized later. The Si set and the related through-going shear zones were observed to depart from the simple shear by ~15° and were close to the 142 S2 S i simple shear S i S i' S i' S i F S i S i Fig. 7.1 Sketches showing the orientations of faults developed under simple shear in (a) and the linkage of Si by Si', S2 and S 2 * forming faults (F) in (b), (c) and (d). The shear sense in (b), (c) and (d) are the same as in (a). 143 greatest principal cq. Therefore in the automaton, it is assumed that faults propagate along this direction, i.e., 15° from the simple shear and 35° from cq. If stable in-plane growth is controlled by either stress corrosion or diffusion controlled subcritical fault growth, then the growth rate, v, is determined by the stress intensity factor K (Atkinson and Meredith 1987, Costin 1987): v °c K n (7.4) Where n is a constant which depends on which mechanism is controlling growth. For mode II crack (corresponding to a strike-slip fau lt), the stress intensity factor, Kji, is related to crack length L by (Li 1987): Kn = {oQ - o f )(KL)m (7.5) where cr0 is the remote applied stress and ay is the shear resistance on the fault surface. Combining equations (7.4) and (7.5) gives v ~ Z * (7.6) where k=n/2. For diffusion controlled cracking, n is between 2-10; for stress corrosion, n ranges from 20 to 50 (Atkinson and Meredith 1987). In order to choose a value for n appropriate to the growth of natural faults, a range of n values were explored in the automaton. As will be shown below, by requiring that the simulated fault patterns have the same geometry as shear networks observed in nature, the author was able to constrain n ~ 2. Hence, based on the simulations, it appears that faults grow at a rate which is linearly proportional to their length. 144 During a time step, the length of each fault on the automaton array is increased by an arbitrarily chosen 10% (5% is added to each end). As discussed in the previous section, this assumption was also made for the growth of the population of starter fault seeds which control nucleation. 7.2.3 Coalescence of faults There is growing evidence that a major component in the growth of strike slip faults is the coalescence of smaller faults (Segall and Pollard 1983b, Deng et al. 1986, Reches 1988, Wesnousky 1988, Cox and Scholz 1988, Martel 1990). Indeed, the existence of numerous bends, jogs, and pull-apart basins argues for coalescence. In order to introduce coalescence into the automaton, it is necessary to formulate rules for the fault configurations which favor coalescence and the distance over which such coalescence can occur. Geometry of coalescence To guide the formulation of automaton rules for coalescence, the experiments in Chapter 3 are again referred. Figs. 3.1 to 3.3 illustrate that the parallel Si shears coalesce by using Si', S2, S2 ' and T structures. In this right- lateral simple shear stress field, the coalescence through Si’ , S2 1 and T bridge structures cause a right step in the compound faults (Figs. 7.1b and d) while coalescence through S2 causes a left-step (Fig. 7.1c). As can be seen, the right- step induces tensile displacement along the jogs while the left-step causes more shear and sometimes, a little compression (indicated by small upwarps in Fig. 3.3c). Note that there is no evidence that left-stepped faults are linked up by structures other than S2 . These experiments confirm that fault 145 coalescence occurs mostly via tension and sometimes shear. Structures such as Si', S 2, S 2 ' and T fault sets which can be either preexisting or newly developed. Detailed bridging geometry is summarized in Fig. 7.1. Based on the above observations, the following rules are adopted for the geometry of fault coalescence in the computer automaton: Strike-slip faults only link up at dilatational or shear offsets (Fig. 7.2). Fig. 7.2a shows the state of stress near the tip of a strike slip fault in a simple shear field. Note that the fault lies at an angle 0 with respect to the direction of simple shear. Because of this, in addition to the usual compressive, C, and tensile, T, stress concentrations, there is a region marked S in which faults can link by the formation of shear structures. As shown in Fig. 7.2b, if one fault tip is in the T region of another, the two faults will link by conjugate shear or by tensile structures to form a right step. However, as shown in Fig. 7.2c, if the fault tip is in the S region of another, the two faults will link by secondary shear S 2 structures forming a left step. Coalescence is not possible if a fault tip is in the C region of another. The radius of the circle, W , is identified as the "process zone" width which will be discussed in the next section. Coalescence range Statistical study of laboratory generated and natural strike-slip faults indicate that the maximum jump distance between two strike-slip faults, W max, is linearly proportional to the combined fault length Lc (An et al., 1994), i.e.: Wmax=RLc (7.7) where R is a constant suggested to be around 0.1. The combined fault length Lc is the total length of two interacting faults. In order to get the interaction 146 simple shear simple shear (b) simple shear (c) Fig. 7.2 (a) shows stress state in front of a fault tip. The area can be divided into a tensile region T, a shear region S, and a compressive region C. The interaction range of a fault is given by W. (b) and (c) show that coalescence is possible only when the tip of one fault is within either the tensile or the shear region of another fault, respectively. Note that, because of the friction on the fault plane, the trend of a strike-slip fault is not parallel to the simple shear direction, but at an angle of 0 . 147 range (i.e., the maximum jump distance) for each individual fault, the author adopts the concept of process zone derived from laboratory experiments (Freidman et al. 1972, Evans et al. 1977, Cox and Scholz 1988, Bjarnason et al. 1992) and field observations (Simpson 1983, Segall and Pollard 1983b, Granier 1985). A process zone is an intensively damaged region around a crack (or fault) tip. The size P of a process zone has been suggested to be proportional to the fault length L (i.e., P/L=0.1 to 0.01, Scholz et al. 1993). Assuming fault interaction is caused by the contact or overlap of two process zones of faults (Fig. 7.3), then Scholz et al.'s (1993) scaling law leads to the scaling relationship between interaction range W of a fault and its length L: which agrees with Eqn. (7.7) because the combination of (7.8) for any two faults leads to (7.7). The constant R is 0.1 according to observation (An et al. Fig. 7.3 Concept of two strike-slip faults interacting over their process zones. The interaction range W for each fault equals to the process zone size which is proportional to the fault length L. W = RL (7.8) max ■ / <- L •> W *RL W =W+W max 1 2 148 1994) and 0.1 to 0.01 according to theoretical prediction (Scholz et al. 1993). Eqn. (7.8) is used to specify the coalescence range in the computer automaton. The parameter R is determined by comparing this simulations with experimental and natural fault patterns. 7.2.4 Summary of automaton rules The rules which govern the nucleation, growth, and coalescence of faults on the automaton array can be summarized as follows: a). Nucleation of faults: Faults nucleate from the growth of starter fault seeds which are assigned a power law size distribution (Eqn. 1). b). Growth of faults: the stable in-plane growth rate v of a fault is assumed to be proportional to the k -th power of fault length L (Eqn. 6). The growth direction is 15° from the simple shear and 75° from o'1 c). Coalescence geometry: Faults only link up at shear and dilatational offsets (Figs. 7.1 and 2). d). Coalescence range: Faults coalesce occurs when process zones of two faults contact or overlap (Fig. 7.3). The interaction range W is assumed to be linearly proportional to fault length L (Eqn. 9). 7.3 Results and discussion Fig. 7.4 shows the evolution of a typical fault pattern. Initially, the fault pattern evolved slowly: it took 18 time-steps to get the pattern shown in Fig. 7.4a. There are few faults in the pattern, and all of them are isolated. This phase is controlled entirely by nucleation and the simple growth law (7.6). 149 18 TIM E — STEPS, B= 4.. R=0 .IO, 500x300 n=2 (a) Fig. 7.4 (continued) 20 TIME-STEPS, B=4., R = 0.10, 500x300 n=2 ( b ) Fig. 7.4 (continued) TIME-STEPS, B=4., R = 0 .’ I0, 500x300 n=2 (c) Fig. 7.4 (continued) :’•! TIM E STEPS, B = 4., R-'O.IO, SOOxSOO (d) Fig. 7.4 (continued) 25 TIME-STEPS, B=4.. R = 0.10, 500x300 (e) Fig. 7.4 (continued) 26 TIME-STEPS, B = 4., R = 0.10. 500x300 n=2 ( f ) Fig. 7.4 (former six pages) Evolution of a fault pattern from time-step 18 in (a) until a though-going shear zone is developed at time step 26 in (f). In this simulation on a 500x300 array, parameters are assumed as B=4, R =0.1, and n =2. Fault interaction began at step 20 when fault density (the ratio of the number of broken cells to intact cells) reached a value of dmin = 0.083 (Fig. 7.4b). The minimum density for interaction, dmin, is a function of the modeling array size, the parameters It and K Beyond dmin, fault growth accelerates, partly because of the increase in the existing fault length, and partly because of the continuous nucleation of new faults (Figs. 7.4c and d). Gradually, fault propagation shifts from growth-dominant to coalescence-dominant, and fault growth become more rapid as coalescing faults cascade (Fig. 7.4e). At the last step (Fig. 7.4f), a critical faull density dc has been reached at which there is at least one fault (here 4) which spans the array. At dc the interaction range of the largest faults become so large and the fault density is so high that coalescence conditions are always satisfied, and one or more through-going shear zones develop. 7.3.2 Comparison with natural fault patterns Morphological sim ilarity. The morphology similarity between the simulated patterns and natural fault patterns is apparent. For example, we can compare strike-slip faults developed in Mojave Desert, Southern California, in Fig. 7.5, with the 156 — r- H 8 ° 1 1 r N t 116 ° _,„•■ • Quaternary Fault " Pre-Quaternary Fault o . ♦ f V 5^ “ "5s" X ^ s -* < t y < - <" / -.\V , \ k ( W . _ ^ 4 ’’ t f \ W S y if'' "Xpse-^A vA X X 'f)V, - ^ " -^ \ js s i.n r > \ s \tt ■ ■V- r v . *X'n V - •* " ■ • v A ** -IL L " v \ x ■ A v x * v k \ J V y ^ *,:. 1 \ " V ' f ' ' p s r \ V •s \ " V X x n m \ L ; ■^•v.:C d M v - \ X, ,-X*~X *•-.. \ f k ' \ x A ,\% ‘ S*'ii4r • u* “ V 'A * * . x - ^ x A v : : ' \% v s t e v V V ' ~ ' ! ^ c v v m . v * ■ % ? - _ . . a >v , " ^ - 5 S ^ \ > s ! , \ x v ^ ^ <„ -**N - i s ^ r *-*. u m '~X -X „ * • % . u-fe— » * . * -. X X r ~ # - 0 20 km I., . 1. j \%"> r ■ Fig. 7.5 Fault pattern of the Mojave Desert in Southern California. Note that this pattern includes all types of faults in the area. AM, Alvord Mountains; AW, Avawatz Mountain; BM, Bristol Mountains; CM, Calico Mountains; CdM, Cady Mountains; CP, Cajon Pass; GM, Granite Mountains; MH, Mud Hills; MM, Marble Mountains; NM, Newberry Mountains; OM, Ord Mountain; PR, Paradise Range; RM, Rodman Mountains; SBM, San Bernardino Mountains. Figure courtesy of Dokka and Travis (1990). simulation in Fig. 7.4f (note however that Fig. 7.5 includes all types of faults, not just strike-slip). The Mojave Desert Block is the site of distributed simpleshear during late Cenozoic time (Dokka and Travis 1990). The faults in the region are discontinuous, with only the Calico-Blackwater fault spanning the entire Mojave Desert, which is similar to a through-going shear zone in our simulation. The larger faults tend to be parallel and have a relatively regular spacing. The traces of the larger faults are complex, involving jogs, bends, overlaps and sometimes, loops. Fault segments show both right- and left-lateral stepping arrangements. All these characters are found in this simulations. We now proceed to more quantitative comparisons between simulated and natural patterns. Fractal fault trace The fault morphology created by this automaton represents an extreme case: a fault trace which is not affected by subsequent displacement and wear. This undisturbed fault morphology permits the author to study fine structure of a fault that may otherwise be obliterated by subsequent tectonic activities. We are interested in measuring the fractal dimension, D, of the largest through-going fault which has been extracted from Fig. 7.4f and plotted in Fig. 7.6a. The "ruler method" was used in which rulers of different lengths were used to measure the length of a fault trace. If the number of rulers used to cover the whole fault trace N(L) is plotted against the lengths of the rulers L in log-log space, the negative slope of the linear portion (if any) of the line gives the fractal dimension D as per the equation N(L)=CL-d (7.9) 158 (a) RULER FRACTAL DIMENSION, Qm 4., R -0 .1 0 , 500x300 0 0 D - 1.0195 U J 0 0 o o C * J L 0 G 2 ( L A G ) (b) Fig. 7.6 (a) is a profile of a through-going shear zone from the simulation shown in Fig. 7.4f. (b) shows the determination of the ruler fractal dimension of the profile in (a). Lag is the ruler length used each time. The fractal dimension is the negative slope of the fitting line. 159 The data in Fig. 7.6b are well fit by a straight line of slope -1.02, indicating that the fault profile generated by the simulation is fractal with a fractal dimension of 1.02. All the fault profiles generated by the automaton have this feature; their fractal dimension D is remarkably stable, at 1.02±0.003, regardless of changes in simulation array size and the initial distribution of starter fault seeds. The fractal dimension of 1.02±0.003 of this simulated fault traces is close to 1.0008-1.0191 obtained for the San Andreas fault by Aviles and Scholz (1987), but a bit lower than 1.1-1.4 for the same fault reported by Okubo and Aki (1987) (probably because they also included local fault branches in their analysis). Fractal dimensions slightly greater than 1 suggest that the fault traces are not exactly linear. Deviation from linearity is caused by coalescence. The undulant fault trace often observed on both vertical and horizontal outcrops in the field may also be explained by this mechanism. Displacement on a fault may smooth out some of the irregularity whereas new coalescence may cause new irregularity. Thus the irregularity of a real fault trace has the components of both older and newer events. Fault length distribution Fault lengths in a simulated pattern can be roughly fit to a power law of the form of Eqn. (7.1), but the fit is generally poor, and the index m in Eqn. (7.1) is much less than that of original starter fault seed distribution. For example, an original m= 2 can drop to less than 1.39 after 26 time-steps of evolution (Fig. 7.7) due largely to fault coalescence. Actually, as a typical example, Fig. 7.7 shows two different trends. Small faults follow a relatively steeper slope while larger faults (generally greater than 25 cell units) have a 160 L E N G T H D IST R IB U T IO N , B= 4., R=0.10, 500x300 in m l= -l.39 L U N in m 2 = —1.09 0 5 10 15 L O G 2(L) Fig. 7.7 Length distribution of the faults in the automaton-generated fault pattern shown in Fig. 7.4f. The original m used in the simulation was 2. 161 flatter slope or sometimes, fluctuate. This is because coalescence had not occurred for small faults, so that they still follow the power law distribution of the starter fault seeds while the coalescence of the larger faults produces a flatter slope. Fault interaction causes rapid growth which outpaces the normal growth rate. New, larger faults emerge by consuming old small faults. The length distribution of natural faults and fractures also follows a power law (Sammis et al. 1992). However the difference in length distribution for natural faults at different scales has not been recognized but should be looked for. The length distribution of segments within some through-going faults is also analyzed (Fig. 7.8a). The distribution has log-normal character in that it has a bell-shaped curve in a semi-log space. Similar distributions are observed for natural faults (Fig. 7.8b and c). Wesnousky (1989) found that fault segment length increases with total fault offset. In this model, zero offset was assumed. Plotting normalized fault length against the normalized displacement, it is found that the automaton follows the same trend as the natural strike-slip faults (Fig. 7.9) in the limit of zero displacement. fault stepping In addition to the average segment length, the occurrence and relative frequency of right and left steps in the automaton-generated pattern can also be compared with natural faults. In the automaton, both right-step and left- step can occur within one fault (Figs. 7.4f and 7.10a), but there are more right- steps than left-steps in this right-lateral shear condition. If considering a lateral jump of 2 mm as a step in Fig. 7.4f, then we can count 21 right-steps and 7 left-steps within the four through-going shear zones in Fig. 7.4f. Those 162 number 50 .0 simulation 4 0 .0 3 0 .0 20.0 10.0 0.0 1 10 100 1000 Length (cell unit) 12.0 San Jacinto 10.0 8.0 6.0 4.0 2.0 100 0.1 10 1 length (km) (b) 4 0 .0 3 5 .0 3 0 .0 San Andreas 5 2 5 .0 E 20.0 3 c 15.0 10.0 5.0 0.0 0.01 0.1 1 10 length (km) 100 (C) Fig. 7.8 Segment length distribution of shear segments in (a) 15 automaton generated through-going shear zones, (b) the San Jacinto fault and (c) the San Andreas fault. 163 D ) c o 3 o O ) c o c 0 £ C D 0 C O 6 > o 1.0 0.1 0.0 I I I I I J I " " I I > ' “ l I ■ I' " 1 o 7 ■Automaton 50 o 3 o o 2 o 6 o 1 From Wesnousky (1989) 1) Newport-Inglewood 2) Whittier-Elsinore 3) San Jacinto 4) Garlock 5) Calaveras 6) San Andreas 7) N.Anatolian i i i i 1 i i i i I i i i i t i i i ■ 0 0.05 0.1 0.15 0.2 0.25 0.3 total displacement/fault length Fig. 7.9 Plot of normalized segment length vs. normalized displacement for natural strike-slip faults (open circles, data from Wesnousky 1989) and for simulated shear zones from this automaton (solid circle). The result from the automaton follows the general trend. characteristics are comparable to those of natural strike-slip faults (Fig. 7.10b, c, d). For example, there are 20 right-steps and 8 left-steps long the San Jacinto fault, 13 right-steps and 9 left-steps along the San Andreas fault from Cholame Valley to Cajon Pass (all counted from USGS miscellaneous investigation series maps 1-675 by Sharp 1972,1-574 by Vedder and Wallace 1970, and 1-553 by Ross 1969), 28 right-steps and 6 left-steps along the North Anatonian fault (counted from Barka and Kadinsky-Code 1988), and 17 right- steps and 11 left-steps along the north western-trending fault systems in Mojave Desert (counted from Fig. 7.5). All these faults are in right-lateral 164 sim u lated f a u lt (a) 10km (c) North Anatonian fault 200km Haiyuan fault 10km (e) San Jacinto fault 20km Fig. 7.10 Arrangements of fault segments in a simulated fault (a) and in natural strike-slip faults (b to e). 165 shear. The statistics shows that there are generally more right-steps than left- steps in a right-lateral strike-slip fault. However, it is interesting to note that if considering only jump distance but not the lateral jump distance, then the counts of right-steps and left-steps are about the same: there are 29 right-steps and 35 left-steps along the four through-going shear zones in Fig. 7.4f. This feature can not be observed for natural strike-slip faults because a left-step as shown in Fig. 7.4f tend to be smoothed out during fault displacement by frictional wearing. Visually a right-step in a right-lateral shear environment appears to be more prominent than does a left-step in the same stress field, assuming the same "jump" distance between the two segment tips. This is because the right-step has a larger lateral jumping component (see Fig. 7.2) Thus it seems that there are more right-steps than left-steps in a right-lateral strike-slip fault. . 7.3.2 Implications Fault interaction range The interaction range in the automaton is controlled by an input variable R, which is the ratio of the radius of the interaction zone to the fault length. It determines the maximum separation over which two faults can coalesce. A large R permits many faults to coalesce, while a small R only allows those which are very close to link up. The model tested different R values, from 0.01 to 0.3, in the simulation. The results are shown in Fig. 7.11. The smallest value of R (0.01) prevented lateral interaction between faults, even when the fault density became so high that nearly all cells were broken (Fig. 7.11a). The result was fault profiles which were much smoother and 166 25 TIME —STEPS, B= 4., R = 0.01, 300x200 Fig. 7.11 Fault patterns generated with (a) R =0.01 and (b) R =0.3. Except for the array size, all the other parameters are the same as those to generate Fig. 7.4 for which R=0.1. Neither (a) nor (b) looks as natural as Fig. 7.4. <r more uniform than those observed in nature. The largest values of R (0.3 in Fig. 7.11b), on the other hand, caused too much lateral linkage and led to fault patterns which were more irregular than those observed in nature (compare Fig. 7.11b with Fig. 7.10. The most realistic patterns were produced when R was between 0.05 and 0.2. It was found that an R value close to 0.1 usually gives fault patterns similar to those observed in nature (Fig. 7.4f). Note that the value of R =0.1 also produces a normalized average segment length which is consistent with natural faults documented by Wesnousky (1989) and experiments (Fig. 7.2a and b). 2.0 1.6 1.1 0.7 0.2 - i — i— i— i— i— i— i— i— i— I— i— i— i— i— |— i— i— i— r - * • —o - D . f d c ....+ -•- m - -O E k ■&- ■3........... ' I I - I I ' I ■ I I I -I I I I I ■ ' ' ■ T ■ . .... 0.05 0.1 0.15 0.2 0.25 0.3 0.35 R Fig. 7.12 Effect of the interaction range coefficient R on the fractal dimension of the main trace, Dft the critical density for the development of a regional fault, dC r and the power law exponent of the final fault length distribution, m. Note that R does not affect Df 169 The effect of R on other fault pattern parameters is shown in Fig. 7.12. Critical fault density dc and average length distribution power m in Eqn.(7.1) decrease with increasing R because, at large R , faults have long-range interaction and thus a through-going shear zone emerges at lower fault density. R also affect the angle 0 between a fault and the shear direction (x- boundary, Fig. 7.2). At large R, 0 is also large (see Fig. 7.11b). The fractal dimensions of the through-going faults are not so sensitive to the change of R. Fault propagation rate Studies of subcritical fault growth show that the rate of fault propagation v is proportional to the n-th power of stress intensity factor K as in Eqn.(7.2), or in terms of fault length, as in Eqn.(7.6). The minimum value of n given by those studies is 2, and the maximum can be as large as 50 (Atkinson and Meredith 1987). The automaton simulations with n >2 all gave fault patterns with a few (usually one) extremely fast-growing faults which spanned the region before others had a chance to develop (Fig. 7.13). It should be noted that in Fig. 7.13, an extremely large value of B =16 was used in Eqn. (7.2) to create the initial pattern of starter faults so that they had little difference in lengths at the beginning. With a smaller B, initial length difference increased, and even fewer smaller faults were developed when the large one spanned the region. Only for n near 2 were realistic fault patterns produced, as shown in Fig. 7.3. According to Eqn. (7.6), n =2 means that the fault propagation rate should be linearly proportional to fault length, i.e.: v = cL (7.10) 170 6 TIME— STEPS, B=16., R=0.10, 600x300 n=4 (a) “ Fig. 7.13 Effect of velocity exponent n on fault patterns, (a) is a fault pattern generated with n =4; (b) is a ~ pattern generated with n =3. Note that for n >2, a smooth region-spanning fault forms before smaller faults can develop and any coalescence can happen. where c is a constant. A small n near 3 is also reported by Main et al. (1993) for compressional crack growth. There are at least three possible explanations for this low value of n. First, the lower crust is ductile. Any brittle deformation in the upper crust would eventually terminate in the lower crust. This is quite different from the conditions used in many laboratory experiments. Since the lower crust resists the brittle shear in the upper crust during coseismic strain release period and drives the creep during interseismic strain accumulation, the overall propagation rate of large strike-slip faults (those which can penetrate the upper crust) slows down. Second, the crust behaves visco-elastic rather than elastic in a long geological time. This means that some of the strain can be accommodated by viscous flow instead of faulting. Third, the crust is heterogeneous. Fault propagation at any scale involves interaction with other faults or defects. Those faults and defects, if not in the right orientation, will arrest the fault propagation. The distribution of starter fault seeds Starter fault seeds are assigned according to a power law distribution (7.2). Several values of B from 2 to 16 (corresponding m from 1 to 4 in Eqn. 7.1) were explored to find one that can be used to generate realistic fault pattern. The fault patterns generated with these B values are shown in Fig. 7.14a to d. Realistic patterns are derived with 2<B <5 (1 <m <2.32). When B >5 (Fig. 7.14d), fault density becomes extremely high (close to one, meaning almost all the cells in an automaton fail), and through-going shear zones trend closer to the simple shear direction (this means the material is too weak to assume significant internal friction). When B <2, too few faults develop in 173 & V o o rO * O o ( £ > o o \\ c£ d \\ CD d c D Xh \ u D % ■ V " CD (6 0 ) % t f > a > o 3 « s C D '- * - < £ * v s S ° is < 3 - * - » O * 'S t ~ c < U 0 ) - C T 3 -k-» o 2 « - * ■ " * * 1 2 _ « s 3 5 a > c /i C O O » s G Q < y ^ X S X 5 * • * - a > , t £ rt 'O lT ) ® * p- co C: < * . C O b D V H ^ H $ £ } <s ' Q I % ^ eo /rt. « - CO £ * • * £ < % & . Q ) d • § . 0> O ■ a ^ O ci < 3 V , O t) < £ < 0 ^ & op w £ ^ 9. P* 9 & I < * * r - 1 . C ~ b D . « ■ ■ » \1A ! > ' \15 Fig. 7.14 (continued) 24 TIME-STEPS, B= 5., R=0.10, 600x300. r?=2 (c) Fig. 7.14 (continued) 19 TIME-STEPS, B = 8 ., R=0.10, 400x200, rr=2 a pattern. The model thus suggests two possibilities: first, for most geological media, the faults may follow a power law distribution with index 1 <m <2.3 in the Eqn. (7.1). The second possibility is that m can be larger than 2.3, but a mobilization limit exists below which faults do not grow. The effect of the starter fault seed distribution on final fault pattern, the density and the fault length distribution is shown in Fig. 7.15. Increasing B leads to an increase in the critical fault density dc (damage) and the fault length distribution exponent m . Again, as with R, fractal dimension D f is not sensitive to B. 2.0 1.6 1.2 0.8 0.4 0.0 n t d c ....+ ••• m i i i i i 1 1 11 i i i 1 1 i i i i 11 i i i i i~i i i 0 — 7-"- ..£..... -O--- f A -o . i i i m r r i i I i i i i I i i i i I i i i i I i i i i I i . i i I i i i i 1.5 2 2.5 3 3.5 4 4.5 5 5.5 B Fig. 7.15 Effects of fault distribution index B on fractal dimension of the main fault trace Df, critical fault density of the final fault pattern dc and length distribution exponent m . Note B does not affect the Df, but raises m and dc as it increases. 178 In the last, the effect of cell array size on the above parameters were examined. The array sizes from 20x10 to 600x400 were explored in the automaton. The effect of changing array size on the critical fault density dc, the fractal dimension Df and the fault length distribution are shown in Fig. 7.16. Generally, increasing the array size causes a slight decrease in dc, but Df remains constant, and m fluctuated slightly. The slight decrease of dc with an increase in array size is understandable, since there is a higher probability 2.0 - - • a - 1.8 ■ + ••• m 1.6 1.4 1.2 : P o - o 1.0 0.8 0.6 0.4 0 100 200 300 400 500 600 700 Array Size Fig. 7.16 Effect of array size (cell number) on fractal dimension of the main fault trace, Dft length distribution exponent m and critical fault density dc . D f again does not change with array size, but d c decreases slightly with increasing array size and m fluctuates slightly. 179 that a larger fault exists and that a faster through-going fault can develop early. 7.4 Summary In summary a computer automaton has been formulated which simulates the following observed properties of natural networks of strike-slip faults: 1. Large strike-slip faults are developed through both growth and coalescence of smaller faults. Coalescence becomes more important when fault density is high. 2. Fault traces are fractal with a fractal dimension of about 1.02. Jog structures within a fault are self-similar over a large range of scales. 3. The length distribution of faults changes with the evolution of the pattern, as the population of large faults increases by consuming small faults. 4. Average segment length in a main region-spanning fault is consistent with those observed by Wesnousky (1989) for natural faults. As stated in the introduction, the automaton was developed to constrain some of the physical parameters which control the evolution of the fault network. By simulating natural fault patterns, as summarized above, the following constraints have been established: 1. The interaction range W is related to fault length L by W=0.1L. 2. Average fault propagation rate is approximately linearly proportional to fault length (v«L). 3. Starter fault seeds follow a power law distribution (Eqn. 1) with power m in the range 1-2.3 (or B=2-5). 180 4. Critical fault density for the formation of a through-going shear zone is positively correlated to starter fault seed distribution index m . 181 Chapter 8 CONCLUSIONS A newly developed simple shear technique, together with computer modeling and theoretical analysis, allows an examination of the nucleation, propagation and interaction of a multitude of shear structures in a broad region. The study revealed that: 1. The development of a strike-slip fault experiences three stages: protofracture, fracture and fault. 2. After nucleation, a shear structure propagates in plane if it constitutes a closed system regarding its environment, propagates out of plane if it constitutes an open system. High confining pressure, high material cohesion and small particle sizes all promote in plane shear propagation. Coarser particle size and excess water help tensile growth. 3. A fracture pattern consists of several generations of conjugate shears as well as tensile fractures. The two primary conjugate shear sets establish a framework within which all the later generations of fractures are developed. Fractures in the most developed primary shear set do not parallel to the applied simple shear direction. 4. Fractures coalesce forming faults and small faults coalesce forming large faults. Faults coalesce by taking advantage of preexisting structures as well as developing new structures. 5. None of the strike-slip faults is parallel to the applied simple shear due to internal friction. The internal friction increases with particle size. 182 6. The releasing steps and restraining steps in a strike-slip fault are developed by at least three mechanisms: (a) coalescence of shear structures (fractures and faults), (b) mutual shearing of shear structures, and (c) mutual avoiding of collinear shear structures. Mechanism (a) creates both releasing and restraining steps while mechanisms (b) and (c) only create restraining steps. Restraining steps created by mechanism (a) permit fault slip while those created by mechanisms (b) and (c) tend to lock fault slip. 7. Displacement along a strike-slip fault leaves two fault walls contacting only at a few points while change all the other parts into pull-apart basins. The contact points work as barriers for fault displacement. Pull-apart basins form along releasing steps and adjacent to low-angle restraining steps. Several basins usually coalesce form a compound basin. 8. Faults rotate with progressive simple shear. 9. Fault interaction range is linearly related to fault length by W=0.1L where if L is a single fault length, W is the interaction range of the single fault. If L is the combined length of two interacting faults, then W is the maximum possible jump distance between the two faults. 10. Fracture propagation rate and displacement are both the linear functions of fracture length. 11. Fault length distribution in a fault pattern follows a power law model. 12. The study suggests that in the shallow crust tensile fractures may be common while at depth, shear structures should prevail. All strike-slip and dip-slip faults are shear structures in a common sense. An important contribution of this study is a new experimental technique for simple shear experiment. Riedel shear technique dominates 183 simple shear experiment for several decades but the shear condition in Riedel-type experiment is not truly simple shear. A shear structure is forced to develop along the simple shear direction when a preexisting "fault" (the boundary between the two driving blocks) moves underneath the sample layer. This might be the reason some believe that strike-slip faults should develop parallel to simple shear direction. In the new technique, true simple shear was realized by fixing one boundary of a sample layer while let the other part moves under gravity body force. A shear structure was generated naturally in this simple shear environment. It turns out that a shear structure is not parallel to the applied simple shear (except minor structure S 2). A unique character of the experiment is that the friction between the sample layer and the base board can be used to simulate the coupling force between the upper brittle crust and the lower ductile crust. The brittle-ductile crust coupling plays an important role in controlling fault movement. The lower ductile crust may drive the deformation of the upper brittle crust during a fault creeping but may drag the movement during an earthquake event. This explains why large strike-slip faults do not run away as predicted by the elastic theory. Another advantage of the experiment is that particle sizes can be controlled to simulate confining pressure effect in the crust. Both fine particle sizes in a granular material and confining pressure in the crust make a material more cohesive. Therefore, a coarse granular material can be used to simulate structure development in the shallow crust where confining pressure is low, while a fine granular material can be used to simulate structure development in the deep crust where confining pressure is high. Moist granular materials also simulate fluid (groundwater and sometimes oil) effect on fault development in the crust. 184 The experiment can be improved in the future by extending it from 2-d to 3-D, by using double layers (upper brittle layer and lower ductile layer), by exploring a variety of other materials with different mechanical properties (e.g., wax, epoxy, resins, plastisin, silicon putty) and by incorporating isostatic effect of the crust in the experiment. The experimental technique can be modified to study tensile and compressional structure development as well. More field geologic investigations and mappings are needed in the future to compare with the laboratory observations. 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Creator An, Linji Y. (author)

Core Title Fault Development Under Simple Shear: Experimental Studies

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Advisor Sammis, Charles G. (committee chair), Aki, Keiiti (committee member), Ershaghi, Iraj (committee member), Lund, Steve P. (committee member)

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